Seminar

  • Timofey Shilkin
  • (Max Planck Institute)
  • On elliptic equations with a singular drift from Morrey spaces
  • Abstract:
  • We investigate weak solutions to the Dirichlet problem for an elliptic equation with a drift b  whose divergence is sign-defined. We assume b belongs to some weak Morrey class which includes in the 3D case, in particular, drifts having a singularity along the axis with the asymptotic  c/r, where r  is the distance to the axis. The problem under consideration is motivated by some questions arising in the theory of axially symmetric solutions to the Navier-Stokes equations. We present results on existence, uniqueness and local properties of weak solutions to this problem as well as its relation to the Navier-Stokes theory. Based on a joint work with M. Chernobai.
  • 23.05.23   10:15

  • Yoshihiro Shibata
  • (Waseda University)
  • Local and global well-posedness of free boundary problem for the Navier-Stokes equations in exterior domains
  • Abstract:
  • See the attached lecture notes.
  •                    09:00

  • Yoshihiro Shibata
  • (Waseda University)
  • Maximal L_p-L_q theory for the Stokes equations with free boundary conditions
  • Abstract:
  • See the attached lecture notes.
  • 16.05.23   08:30

  • Yoshihiro Shibata
  • (Waseda University)
  • Introduction to free boundary problem for the Navier-Stokes equations and R-solver approach to this problem
  • Abstract:
  • See the attached lecture notes.
  • 09.05.23   09:00

  • Florian Oschmann
  • (Institute of Mathematics, CAS)
  • Some insights in homogenization of compressible Navier-Stokes equations
  • Abstract:
  • We investigate the homogenization of the two- and three-dimensional unsteady compressible Navier-Stokes equations, and show (i) the convergence result for the 2D case, and (ii) a better lower bound for the adiabatic exponent occurring in the pressure law for the 3D case. The latter will be covered with two different proofs. Lastly, we briefly discuss the optimality of the achieved bounds for the adiabatic exponent in terms of the underlying space-time dimension. This is joint work with Šárka Nečasová (CAS) and Milan Pokorný (Charles University).
  • 02.05.23   09:00

  • Maja Szlenk
  • (University of Warsaw)
  • A multifluid model with chemically reacting components - construction of weak solutions
  • Abstract:
  • We investigate the existence of weak solutions to the multi-component system, consisting of compressible chemically reacting components, coupled with the Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reaction and assume the nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum.

    It is shown that there exists a global weak solution, satisfying the $L^infty$ bounds for all the components. Moreover, despite the lack of the regularity on the gradients, we obtain strong compactness of densities in $L^2$. The applied method captures the properties of the models of high generality, which admit an arbitrary number of components. Furthermore, the framework we develop can handle models that contain both diffusing and non-diffusing elements.
  • 25.04.23   09:00

  • Petr Kaplický
  • (Charles University)
  • Stokes problems with dynamic boundary conditions
  • Abstract:
  • We show maximal regularity in time of solutions to the evolutionary Stokes problem with dynamic boundary condition in the case that the underlying space is Hilbert space.
  • 18.04.23   09:00

  • Eduard Feireisl
  • (Institute of Mathematics, CAS)
  • Compressible MHD as a dissipative system
  • Abstract:
  • We show that the compressible MHD system admits a bounded absorbing set in the energy ``norm'' as long as the open boundary conditions are imposed. In addition, the trajectories are precompact in a suitable topology. If this is the case, there is a compact global attractor as well as statistical stationary solutions supported by individual trajectories of weak solutions.
  • 11.04.23   09:00

  • Anna Abbatiello
  • (University of L'Aquila)
  • On the stability of incompressible heat conducting non-newtonian fluids
  • Abstract:
  • See the attached file.
  • 04.04.23   11:30

  • Nilasis Chaudhuri
  • (Imperial College)
  • Construction of the weak solutions to the barotropic Navier-Stokes system compatible with Kolmogorov compactness criterion
  • Abstract:
  • We prove the existence of the weak solutions to the compressible Navier--Stokes system with barotropic pressure $p(varrho)=varrho^gamma$ for $gammageq 9/5$ in three space dimensions. In the approximation scheme we use more direct truncation and regularisation of nonlinear terms and the pressure instead of the classical regularization of the continuity equation (based on the viscosity approximation $ep Delta varrho$). This scheme is compatible with the Kolmogorov (Bresch-Jabin) compactness criterion for the density. We revisit this criterion and prove that it can be applied in our approximation at any level.
  •                    10:15

  • Eva Fašangová
  • (TU Dresden)
  • The fractional Laplace operator (an introduction)
  • Abstract:
  • We motivate the heat equation with a fully discrete model (lattice gas cellular automaton), the classical as well as the one with the  fractional Laplace operator. We discuss the mathematical setting for the fractional case: the space (fractional Sobolev), the operator, the energy, boundary conditions, weak solution.
  •                    09:00

  • Thomas Eiter
  • (WIAS Berlin)
  • The concept of energy-variational solutions for hyperbolic conservation laws
  • Abstract:
  • We consider the notion of energy-variational solutions for hyperbolic conservation laws. This novel solvability concept is obtained by enriching the variational formulation by the weighted difference between the mechanical energy and an auxiliary variable representing the turbulent energy. If the weight is chosen suitably, an existence result for a general class of conservation laws can be derived via a time-discretization scheme based on a sequential minimization and, in particular, without a spatial regularization. The solution concept comes along with favorable properties like a weak-strong uniqueness principle and the convexity of solution sets. Moreover, for the compressible and incompressible Euler equations, energy-variational solutions can be identified with dissipative weak solutions.
  • 28.03.23   09:00

  • Antonín Češík
  • (Charles University)
  • Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision
  • Abstract:
  • The talk discusses existence theory for collisions of (visco-)elastic bulk solids which are undergoing inertial evolution. In particular, our approach for contact is based only on the assumption of non-interpenetration of matter. Most other theories for contact of elastic solids include some phenomenological assumptions, which we do not need in our approach.

    We are able to show existence of weak solutions including contact with an obstacle or with the solid itself, for arbitrarily large times and large deformations. Furthermore, our construction includes a characterization of the contact force which obeys conservation of momentum and an energy balance. This contact force is a vector-valued surface measure acting in the normal direction, and is constructed as a consequence of the non-interpenetration of matter.

    This is a joint work with Giovanni Gravina and Malte Kampschulte.
  • 21.03.23   09:00

  • Volodymyr Mikhailets
  • (Institute of Mathematics, CAS)
  • One-dimensional differential operators with distributions in coefficients
  • Abstract:
  • Some classes of linear ordinary differential operators with strongly singular coefficients are studied in the talk. These operators are introduced as quasi-differential according to Shin-Zettl. Their domains may not contain non-zero smooth functions. The case of self-adjoint Schrödinger and Hill operators on the line is investigated in more detail.
  • 14.03.23   09:00

  • Aneta Wróblewska-Kamińska
  • (Institute of Mathematics, Polish Academy of Sciences)
  • Relaxation limit of hydrodynamic models
  • Abstract:
  • We will show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We plan to discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We will deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we will mention how to show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.
    This is a joint result with Jose Carrillo and Yingping Peng.
  • 07.03.23   09:00

  • Ivan Gudoshnikov
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Elastoplasticity with softening in spring network models: a state-dependent sweeping process approach
  • Abstract:
  • Softening plasticity and Gurson model of damage in particular lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete failure of a material. Moreover, spatially continuous models with softening suffer from localization of strains and stresses to measure-zero submanifolds. We formulate a problem of quasistatic evolution of elasto-plastic spring networks (Lattice Spring Models) with a plastic flow rule which describes linear hardening, linear softening and perfectly plastic springs in a uniform manner. The fundamental kinematic and static characteristics of the network are described by the rigidity theory and structural mechanics.
    To solve the evolution problem we convert it to a type of a differential quasi-variational inequality known as the state-dependent sweeping process. We prove the existence of solution to the associated time-stepping problem (implicit catch-up algorithm), and the estimates we obtain imply the existence of a solution to the (time-continuous) sweeping process. Using numerical simulations of regular grid-shaped networks with softening we demonstrate the development of non-symmetric shear bands. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions, appearing in a bifurcation which happens when the parameters of the networks continuously change from hardening through perfect plasticity to softening.
  • 28.02.23   09:00

  • Sourav Mitra
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Existence of weak solutions for a compressible multi-component fluid structure interaction problem
  • Abstract:
  • See the attached file.
  • 21.02.23   09:00

  • Matthieu Cadiot
  • (McGill University, Canada)
  • Rigorous Computation of Solutions of Semi-Linear Partial Differential Equations on Unbounded Domains Via Spectral Methods
  • Abstract:
  • In recent years, rigorous numerics have become a major tool to prove solutions of Partial Differential Equations (PDEs). However, when the equation is set on an unbounded domain of $mathbb{R}^m$ (where $m ge 1$), only a few results have been obtained so far. In this talk, I will present a general method to rigorously prove strong solutions to a large class of nonlinear PDEs in a Hilbert space $H^l subset H^s(mathbb{R}^m)$ ($sgeq 1$) via computer-assisted proofs.  We first introduce a method to rigorously compute an upper bound for the norm of the inverse of the linearization of PDE operators. The method is purely spectral and the constants are determined through Fourier analysis. Then using a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions to PDEs. I will illustrate the method using the special case of the Kawahara equation (fourth order KdV equation) and, as an application, I will present some computer-assisted proofs of localized patterns for the 2D Swift-Hohenberg equation.
  • 14.02.23   09:00

  • Tomasz Debiec
  • (University of Warsaw)
  • On the incompressible limit for some tissue growth models
  • Abstract:
  • I will discuss some approaches to mathematical modelling of living tissues, with application to tumour growth. In particular, I will describe recent results on to the incompressible limit of a compressible model, which builds a bridge between density-based description and a geometric free-boundary problem by passing to the singular limit in the pressure law.

    The talk is divided in two parts. First, I discuss the rate of convergence of solutions of a general class of nonlinear diffusion equations of porous medium type to solutions of a Hele-Shaw-type problem. Then, I shall present a two-species tissue growth model — the main novelty here is the coupling of both species through the so-called Brinkman law which is typically used in the context of visco-elastic media, where the velocity field is linked to the total population pressure via an elliptic equation.
  • 13.12.22   09:00

  • Helmut Abels
  • (University of Regensburg)
  • Regularity and Convergence to Equilibrium for a Navier-Stokes-Cahn-Hilliard System with Unmatched Densities
  • Abstract:
  • We study the initial-boundary value problem for an incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory  for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as time tends to infinity. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity belonging only to the Leray-Hopf class, the energy dissipation of the system, the separation property for large times, a weak strong uniqueness type result, and the Lojasiewicz-Simon inequality.
  • 15.11.22   09:00

  • Elek Csobo
  • (University of Innsbruck)
  • On blowup for the supercritical quadratic wave equation
  • Abstract:
  • See the attached file.
  • 08.11.22   10:15

  • Amjad Tuffaha
  • (American University of Sharjah)
  • On the well-posedness of an inviscid fluid-structure interaction model
  • Abstract:
  • We consider the Euler equations on a domain with free moving interface. The motion of the interface is governed by a 4th order linear Euler-Bernoulli beam equation. The fluid structure interaction  dynamics are realized through normal velocity matching of the fluid and the structure in addition to the aerodynamic forcing due to the fluid pressure.
    We derive a-priori estimates and construct local-in-time solutions to the system in the Sobolev space H^r, with r>5/2. We also establish uniqueness in the Sobolev space H^r with r>3. An important consequence of the existence theorem is that the Taylor-Rayleigh instability does not occur. This is joint work with Igor Kukavica.
  •                    09:00

  • Viktor Hruška
  • (Czech Technical University in Prague)
  • Problems of linearized Navier-Stokes equations in frequency domain
  • Abstract:
  • For aeroacoustics applications, it is very tempting to work with linearized equations in frequency domain. Not only are the solutions simpler in overall, but also some variables are defined solely in the frequency domain (such as impedance and related quantities). In quiescent media, the frequency domain calculations enjoy well-deserved popularity. However, great caution must be taken when applying the same mathematical steps to linearized Navier-Stokes equations, although technically there is no apparent difficulty. The talk will present a specific case of the method failure: despite the fact that the acoustic quantities are indeed small, the hydrodynamics cannot be governed by the linearized equations. The final part of the talk will be a discussion of some papers that use the linearized equations.
  • 01.11.22   10:15

  • Richard Höfer
  • (Institut de Mathématiques de Jussieu)
  • On the derivation of viscoelastic models for Brownian suspensions
  • Abstract:
  • We consider effective properties of suspensions of inertialess, rigid, anisotropic, Brownian particles in Stokes flows. Recent years have seen tremendous progress regarding the rigorous justification of effective fluid equations for non-Brownian suspensions, where the complex fluid can be described in terms of an effective viscosity. In contrast to this (quasi-)Newtonian behavior, anisotropic Brownian particles cause an additional elastic stress on the fluid. A  rigorous derivation of such visco-elastic systems starting from particle models is completely missing so far. In this talk I will present first results in this direction starting from simplified microscopic models where the particles evolve only due to rotational Brownian motion and cause a Brownian torque on the fluid. In the limit of infinitely many small particles with vanishing particle volume fraction, we rigorously obtain an elastic stress on the fluid in terms of the particle density that is given as the solution to an (in-)stationary Fokker-Planck equation.
    Joint work with Marta Leocata (LUISS Rome) and Amina Mecherbet (Université Paris Cité)
  •                    09:00

  • Nilasis Chaudhuri
  • (Imperial College)
  • Analysis of generalized Aw-Rascle type model
  • Abstract:
  • In this talk we consider the multidimensional generalization of the Aw-Rascle system for vehicular traffic. For a large class of initial data and the periodic boundary conditions, we prove the existence of a global-in-time measure-valued solution. Moreover, using the relative energy technique, we show a weak-strong uniqueness result. Next, we analyse the similar generalization in one dimensional setting by considering the offset function is a gradient of a singular function of the density and the resulting system of PDEs can be used to model traffic or suspension flows with the maximal packing constraint taken into account. We study the so-called 'hard congestion limit' and show the convergence of solutions towards a weak solution of a hybrid free-congested system.
  • 25.10.22   09:00

  • John Sebastian Simon
  • (Kanazawa University)
  • Convergence of shape design solutions for the Navier-Stokes equations
  • Abstract:
  • We investigate the behavior of dynamic shape design problems for fluid flow at large time horizon. In particular, we shall compare the solutions of a dynamic shape optimization problem with that of a stationary problem and show that the solution of the former converges to that of the latter. The convergence of domains is based on the $L^infty$-topology of their corresponding characteristic functions which is closed under the set of domains satisfying the cone property. Lastly, a numerical example is provided to show the occurrence of such convergence.
  • 11.10.22   09:00

  • Florian Oschmann
  • (Institute of Mathematics, CAS)
  • An unexpected term for the Oberbeck--Boussinesq approximation
  • Abstract:
  • The Rayleigh-B'enard convection problem deals with the motion of a compressible fluid in a tunnel heated from below and cooled from above. In this context, the so-called Boussinesq relation is used, claiming that the density deviation from a constant reference value is a linear function of the temperature. These density and temperature deviations then satisfy the so-called Oberbeck-Boussinesq equations. The rigorous derivation of this system from the full compressible Navier-Stokes-Fourier system was done by Feireisl and Novotn'y for conservative boundary conditions on the fluid's velocity and temperature. In this talk, we investigate the derivation for Dirichlet boundary conditions, and show that differently to the case of conservative boundary conditions, the limiting system contains an unexpected non-local temperature term. This is joint work with Peter Bella (TU Dortmund) and Eduard Feireisl (CAS).
  • 04.10.22   09:00

  • Srđan Trifunović
  • (University of Novi Sad)
  • Global existence of weak solutions in nonlinear 3D thermoelasticity
  • Abstract:
  • See the attached file.
  • 14.06.22   09:00

  • Yassine Tahraoui
  • (NOVA University Lisbon)
  • On deterministic and stochastic obstacle problems
  • Abstract:
  • See the attached file.
  • 31.05.22   09:00

  • Buddhika Priyasad
  • (Charles University)
  • Uniform boundary stabilization of the 3D- Navier-Stokes Equations and of 2D and 3D Boussinesq system by Finite dimensional localized boundary feedback controllers in Besov spaces of low regularity
  • Abstract:
  • In this talk, I present two stabilization problems of fluid equations, namely the Navier-Stokes Equations and the Boussinesq System, both in d = 2,3 setting. For the Navier Stokes problem, we use two localized controls {v, u} where the boundary control v localized on a small portion of the boundary and the interior control u localized on an arbitrarily small collar supported on the same boundary portion. For the Boussinesq problem, we use two localized controls {v, u} where v acting on the thermal equation as a localized boundary control and u acting as a localized interior control for the fluid equation. The initial conditions for both systems are taken of low regularity. We then seek to uniformly stabilize both systems in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair {v, u}. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L^3 for d=3 ) and a  corresponding Besov space for the thermal component. Unique continuation inverse theorems for suitably over-determined adjoint static problems play a
    critical role in the constructive solution.
  • 18.05.22   10:15

  • Zihui He
  • (University of Bielefeld)
  • On some two-dimensional incompressible inhomogeneous viscous fluid flows
  • Abstract:
  • In this talk, we will present some existence, uniqueness and regularity results for the motion of two-dimensional incompressible inhomogeneous viscous fluid flows in presence of a density-/temperature-dependent viscosity coefficient.

    Firstly, we will discuss the boundary value problem for the stationary Navier-Stokes equation, where the viscosity coefficient is density-dependent. We will give some explicit solutions with piecewise constant viscosity coefficients, where some regularity and irregularity results will be considered.

    We will also discuss the initial value problem for the evolutionary Boussinesq equation, which is a nonlinear coupling between a heat equation and a Navier-Stokes type of equation. In this case, the viscosity coefficient is temperature-dependent.

    This talk is based on joint work with Xian Liao (KIT).
  •                    09:00

  • Aleksandr Murach
  • (NAS of Ukraine, Institute of Mathematics)
  • Parabolic boundary-value problems in generalized Sobolev spaces
  • Abstract:
  • See the attached file.
  • 03.05.22   09:00

  • Huanyao Wen
  • (South China University of Technology)
  • Global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions
  • Abstract:
  • We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluids and the momentum equations for the mixture. This model can be derived from a generic compressible two-fluid model with equal velocities and from a scaling limit of the Vlasov-Fokker-Planck/compressible Navier-Stokes equations. Under some assumptions on the initial data which can be discontinuous, unbounded and large, we show existence of global weak solutions with finite energy. The main difference compared with previous works on the same model, is that transition to each single-phase flow is allowed without any domination conditions of densities.

    Reference: H. Wen, On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions. Calc. Var. (2021) 60:158.
  • 26.04.22   09:00

  • María Ángeles Rodríguez-Bellido
  • (University of Sevilla)
  • Results for a bilinear control problem associated to a repulsive chemotaxis model
  • Abstract:
  • Chemotaxis is understood as the biological process of the movement of living organisms in response to a chemical stimulus which can be given towards a higher (attractive) or lower (repulsive) concentration of a chemical substance. At the same time, the presence of living organisms can produce or consume chemical substance.
    In this talk, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a 2D and 3D models. The existence of a global optimal solution with bilinear control is analyzed. First-order optimality conditions for local optimal solutions are derived by using a Lagrange multiplier theorem.

    References:

    [1] Guillén-González, F.; Mallea-Zepeda, E.; Rodríguez-Bellido, M. A.
    Optimal bilinear control problem related to a chemo-repulsion system in 2D domains.
    ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 29, 21 pp.

    [2] Guillen-Gonzalez, F.; Mallea-Zepeda, E.; Rodriguez-Bellido, M. A.
    A regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem.
    SIAM J. Control Optim. 58 (2020), no. 3, 1457–1490.
  • 19.04.22   09:00

  • Justyna Ogorzaly
  • (Jagiellonian University, Krakow)
  • Variational-Hemivariational Inequalities with Applications to Contact Mechanics
  • Abstract:
  • We will present the existence and uniqueness results for the special classes of nonlinear variational-hemivariational inequalities. Then, we will consider concrete contact problems and we will show how these problems lead to the different type of variational-hemivariational inequalities.
  • 05.04.22   09:00

  • Aneta Wróblewska-Kamińska
  • (Institute of Mathematics, Polish Academy of Sciences)
  • Two-phase compressible/incompressible Navier-Stokes system with inflow-outflow boundary conditions
  • Abstract:
  • I will show proof of the existence of a weak solution to the compressible Navier-Stokes system with singular pressure that explodes when density achieves its congestion level. This is a quantity whose initial value evolves according to the transport equation. We then prove that the “stiff pressure" limit gives rise to the two-phase compressible/incompressible system with congestion constraint describing the free interface. We prescribe the velocity at the boundary and the value of density at the inflow part of the boundary of a general bounded C2 domain. For the positive velocity flux, there are no restrictions on the size of the boundary conditions, while for the zero flux, a certain smallness is required for the last limit passage. This result is based on a work with Milan Pokorný and Ewelina Zatorska.
    References:
    M. Pokorný, A. Wróblewska-Kami?ska, E. Zatorska. Two-phase compressible/incompressible Navier–Stokes system with inflow-outflow boundary conditions. arXiv:2202.03557, 2022.
  • 22.03.22   09:00

  • Colette Guillopé
  • (Paris-East Créteil University)
  • About a 1D Green-Naghdi model with vorticity and surface tension for surface waves
  • Abstract:
  • The Green-Naghdi model is currently the most well-known model used for numerical simulations of waterfront streams, even in setups that incorporate vanishing depth (at the shoreline) and wave breaking. Regardless of their many favorable circumstances, the Green-Naghdi equations specially take into consideration neglected rotational effects, which are significant for wind-driven waves, waves riding upon a sheared current, waves near a ship, or tsunami waves approaching a shore. The Green-Naghdi system is first rewritten as an equivalent system by using an adequate change of unknowns. We show that solutions to the model here considered, with voracity and surface tension, may be obtained by a standard Picard iterative scheme. No loss of regularity is involved with respect to the initial data. Moreover solutions exist at the same level of regularity as for 1st order hyperbolic symmetric systems, i.e. with a regularity in Sobolev spaces of order s > 3/2.
  • 15.03.22   10:15

  • Boris Muha
  • (University of Zagreb)
  • Poroelasticity Interacting with Stokes Flow
  • Abstract:
  • We consider the interaction between an incompressible, viscous fluid modeled by the dynamic Stokes equation and a multilayered poroelastic structure which consists of a thin, linear, poroelastic plate layer (in direct contact with the free Stokes flow) and a thick Biot layer. The fluid flow and the elastodynamics of the multilayered poroelastic structure are fully coupled across a fixed interface through physical coupling conditions (including the Beavers-Joseph-Saffman condition), which present mathematical challenges related to the regularity of associated velocity traces. We prove existence of weak solutions to this fluid-structure interaction problem with either (i) a linear, dynamic Biot model, or (ii) a nonlinear quasi-static Biot component, where the permeability is a nonlinear function of the fluid content (as motivated by biological applications). The proof is based on constructing approximate solutions through Rothe’s method, and using energy methods and a version of Aubin-Lions compactness lemma (in the nonlinear case) to recover the weak solution as the limit of approximate subsequences. We also provide uniqueness criteria and show that constructed weak solutions are indeed strong solutions to the coupled problem if one assumes additional regularity.
    The presented results are joint work with L. Bociu, S. Čani? and J. Webster.
  •                    09:00

  • Jean-Baptiste Clément
  • (Czech Technical University)
  • Adaptive solution strategy for Richards' equation based on Discontinuous Galerkin methods and mesh refinement
  • Abstract:
  • Richards' equation describes flows in variably saturated porous media. Its solution is challenging since it is a parabolic equation with nonlinearities and degeneracies. In particular, many real-life problems are demanding because they can involve steep/heterogeneous hydraulics properties, dynamic  boundary conditions or moving sharp wetting fronts. In this regard, the aim is to design a robust and efficient numerical method to solve Richards’ equation. Towards this direction, the work presented here deals with Discontinuous Galerkin methods which are very flexible discretization schemes. They are combined with BDF methods to get high-order solutions. Built upon these desirable features, an adaptive mesh refinement strategy is proposed to improve Richards’ equation simulations. Examples such as the impoundment of a multi-material dam or the groundwater dynamics of sandy beaches illustrate the abilities of the approach.
  • 01.03.22   09:00

  • Ivan Gudoshnikov
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Sweeping process and its stability with applications to lattices of elasto-plastic springs
  • Abstract:
  • Moreau's sweeping process is a class of non-smooth evolution problems invented to handle one-sided constraints in natural processes involving e.g. elastoplasticity, friction and thresholds in electicity and electomagnetism. The sweeping process can be viewed as a geometric generalization of hysteresis models. I will discuss its asymptotic properties, especially focusing on the case of a periodic input, as it leads to periodic outputs forming an attracting set.
    Another focus will be the stress analysis of lattices of elasto-plastic springs via a finite-dimensional sweeping process (with illustrative examples). The mentioned asymptotic properties lead to nice conclusions about stress trajectories in the lattice models.
    This is a joint project with Oleg Makarenkov, Dmitry Rachinskiy (University of Texas at Dallas) and Yang Jiao (Arizona State University).
  • 22.02.22   09:00

  • Yong Lu
  • (Nanjing University)
  • Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable
  • Abstract:
  • We prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which is slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to  2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of constant state nor of small energy.
  • 04.01.22   09:00

  • Maja Szlenk
  • (University of Warsaw)
  • Uniqueness of weak solutions for the Stokes system for compressible fluids with general pressure
  • Abstract:
  • We prove existence and uniqueness of global in time weak solutions for the Stokes system for compressible fluids with a general, non-monotone pressure. We construct the solution at the level of Lagrangian formulation and then define the transformation to the original Eulerian coordinates. For a nonnegative and bounded initial density, the solution is nonnegative for all $t>0$ as well and belongs to $L^infty([0,infty)timesmathbb{T}^d)$. A key point of our considerations is the uniqueness of such transformation. Since the velocity might not be Lipschitz continuous, we develop a method which relies on the results of Crippa & De Lellis, concerning regular Lagriangian flows. The uniqueness is obtained thanks to the application of a certain weighted flow and detail analysis based on the properties of the $BMO$ space.
  • 30.11.21   09:00

  • Clara Patriarca
  • (Politecnico di Milano)
  • Existence and uniqueness result for a fluid-structure-interaction evolution problem in an unbounded 2D channel
  • Abstract:
  • In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a Poiseuille flow profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent the obstacle from going excessively far from the equilibrium position and colliding with the boundary of the channel, we introduce a strong force in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution.
  • 23.11.21   09:00

  • Sourav Mitra
  • (University of Würzburg)
  • A control problem of a linear compressible fluid-structure interaction model
  • Abstract:
  • I will talk about a result on the controllability of a compressible FSI model where the structure is located at a part of the fluid boundary. I will first introduce the notion of control and explain the tools to prove the controllability of a linear PDE. In the next part of the talk I will introduce the FSI model under consideration and corresponding linearization. Finally I will speak about the control of the linearized FSI problem and outline a proof.
  • 16.11.21   09:00

  • Srđan Trifunović
  • (University of Novi Sad)
  • On the fluid-structure interaction problem with heat exchange
  • Abstract:
  • Here, I will talk about a nonlinear interaction problem between a thermoelastic shell and a heat-conducting fluid. The shell is governed by linear  thermoelasticity equations and constitutes a time-dependent domain which is filled with a fluid governed by the full Navier-Stokes-Fourier system. The fluid and the shell are fully coupled, giving rise to a new previously unstudied interaction problem involving heat exchange. The existence of a weak solution for this problem is obtained by combining three approximation techniques - decoupling, penalization and domain extension for fluid.
    This talk is based on a joint work with Václav Mácha, Boris Muha, Šárka Nečasová and Arnab Roy.
  • 09.11.21   09:00

  • Ana Radošević
  • (University of Zagreb)
  • On the regularity of weak solutions to the fluid-rigid body interaction problem
  • Abstract:
  • The fluid-structure interaction (FSI) systems are multi-physics systems that include a fluid and solid component. They are everyday phenomena with a wide range of applications. The simplest model for the structure is a rigid body. We study a nonlinear moving boundary fluid-structure interaction problem where the fluid flow is governed by 3D Navier-Stokes equations, and the structure is a rigid body described by a system of ordinary differential equations called Euler equations for the rigid body. Our goal is to show that a weak solution that additionally satisfy Prodi-Serrin condition is smooth on the interval of its existence, which is a generalization of the well-known regularity result for the Navier-Stokes equations. This is a joint work with Boris Muha and Šárka Nečasová.
  • 02.11.21   09:00

  • Danica Basarić
  • (Institute of Mathematics, CAS)
  • Existence of weak solutions for models of general compressible viscous fluids with linear pressure
  • Abstract:
  • In this talk we will focus on the existence of weak solutions for a system describing a general compressible viscous fluid in the case of the pressure being a linear function of the density and the viscous stress tensor being a non-linear function of the symmetric velocity gradient. More precisely, we will first prove the existence of dissipative solutions and study under which conditions it is possible to guarantee the existence of weak solutions.
  • 26.10.21   09:00

  • Emil Skříšovský
  • (Charles University)
  • Evolutionary compressible Navier-Stokes-Fourier system in two space dimensions with adiabatic exponent almost one
  • Abstract:
  • In this talk we consider the full Navier-Stokes-Fourier system and present the proof of the existence of a weak solution in two space dimensions for the pressure law given by $p(varrho,theta) sim varrhotheta + varrho log^alpha(1+varrho)+ theta^4$, which can be viewed as a close approximation of the pressure law for ideal gas $p(varrho,theta) sim varrhotheta$. The weak solutions with entropy inequality and total energy balance are considered and the existence of this type of weak solutions without any restriction on the size of the initial conditions or the right-hand sides is shown provided $alpha > frac{17+sqrt{417}}{16}cong 2.34$.
  • 19.10.21   09:00

  • Milan Pokorný
  • (Charles University)
  • Steady compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions for the temperature
  • Abstract:
  • Based on recent result by Chaudhuri and Feireisl for the evolutionary NSF system we present the proof of existence of weak (and variational entropy) solutions to the steady version with Dirichlet boundary conditions for the temperature. The formulation is based, similarly as in the evolutionary case, on a version of ballistic energy inequality which allows to obtain a priori estimates for the temperature and velocity.
  • 05.10.21   09:00

  • Peter Bella
  • (TU Dortmund)
  • Regularity for degenerate elliptic equations
  • Abstract:
  • I discuss local regularity properties of solutions of linear non-uniformly elliptic equations with non-constant coefficients. Assuming certain integrability conditions on the ellipticity of the coefficient field, we obtain local boundedness of weak solutions and corresponding Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results in the literature (Trudinger). I will also discuss analogous result for the time-independent parabolic equations as well as application to study of the variational integrals with differential (p,q) growth.
  • 08.09.21   10:10

  • Florian Oschmann
  • (TU Dortmund)
  • Inverse of divergence and homogenization of compressible Navier-Stokes equations in randomly perforated domains
  • Abstract:
  • In homogenization of compressible Navier-Stokes equations, an inverse of the divergence operator (called Bogivskiu{i} operator) is crucial to obtain a-priori bounds for the velocity and density independent of the perforation. Such Bogovskiu{i} operators and bounds are well known in the case of periodically arranged holes with fixed diameter, where the mutual distance is of order $varepsilon>0$ and the radii scale like $varepsilon^alpha$ for some $alpha>3$. We generalize these results to the case of randomly distributed holes with random radii and give applications to the homogenization of the Navier-Stokes(-Fourier) equations in such randomly perforated domains.
  •                    09:00

  • Nilasis Chaudhuri
  • (TU Berlin)
  • Convergence of consistent approximations to the complete compressible Euler system
  • Abstract:
  • The aim of the talk is to discuss a result on the weak limit of a `consistent approximation scheme' to the compressible complete Euler system in the full space $ mathbb{R}^d,; d=2,3 $. The main result states that if a weak limit of the consistent approximation scheme is a weak solution of the system, then the approximate solutions converge locally strongly (or at least almost everywhere) in suitable norms under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general and includes the vanishing viscosity and heat conductivity limit. In particular, they do not have to satisfy the minimal principle for entropy.
  • 15.06.21   09:00

  • Ewelina Zatorska
  • (Imperial College London)
  • On the existence of solutions to the two fluids systems
  • Abstract:
  • We prove the existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions–Feireisl approach. Existence, uniqueness and stability of global weak solutions to this system are obtained with arbitrarily large initial data. Making use of the uniform-in-time bounds for the densities from above and below, exponential decay of weak solution to the unique steady state is obtained without any smallness restriction to the size of the initial data. In particular, our results show that degeneration to single-fluid motion will not occur as long as in the initial distribution both components are present at every point.
    The results are based on joint papers with D. Bresch, P. Mucha, Y. Li and Y. Sun.
  • 25.05.21   10:00

  • Frank Merle
  • (Université de Cergy-Pontoise)
  • On the implosion of a three dimensional compressible fluid
  • Abstract:
  • We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. Two essential steps of analysis are the existence of very regular self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).
  • 18.05.21   09:00

  • Yoshihiro Shibata
  • (Waseda University)
  • R-solver, Maximal Regularity, and Mathematical Fluid Dynamics
  • Abstract:
  • Maximal Regularity is an important tool to show the existence of strong solutions of quasi-linear system of parabolic equations, for example free boundary problems for the Navier-Stokes equations. In this lecture, I will talk about a systematic approach for the Lp maximal regularity by using the R-solver. This approach is quite useful to control the high frequency part of solutions to the linearized problem, and so we can prove the local well-posedness for the dynamical equations appearing in the mathematical fluid dynamics. But, to prove the global well-posedness at least for small initial data, we have to control the low frequency part. To do this I use so called Lp-Lq decay estimate for the semigroup group associated with the linearized problem, like Stokes equations. In this talk, I will present how to prove the maximal regularity by using an R-solver and how to control the low frequency part by using Lp-Lq estimates to prove the global wellposedness in some concrete example, like equations describing the compressible viscous fluid flow.
  • 11.05.21   09:00

  • Paolo Maremonti
  • (Universitŕ della Campania Luigi Vanvitelli)
  • On the uniqueness of a suitable weak solution to the Navier-Stokes Cauchy problem
  • Abstract:
  • We are dealing with the Navier-Stokes Cauchy problem. We investigate some results of regularity and uniqueness related to suitable weak solutions. The suitable weak solution notion is meant in the sense introduced by Caffarelli-Kohn-Nirenberg. In paper [1], we recognize that a suitable weak solution enjoys more regularity than Leray-Hopf weak solutions, that allows us to furnish new uniqueness results for the solutions. Actually, we realize two results. The first one is a new sufficient condition on the initial datum u0 for uniqueness. We work on existing suitable weak solution, that is, we do not construct a more regular weak solution corresponding to our initial datum. The second result employs a weaker condition with respect to previous ones (almost $u_0 in  L^2$), but, just for one of the two compared weak solutions, we need a “special” Prodi-Serrin condition. It is “special” as it is local in space.

    References: [1] Crispo F. and Maremonti P., On the uniqueness of a suitable weak solution to the Navier-Stokes Cauchy problem, SN Partial Differential Equations and Applications, to appear.
  • 04.05.21   09:00

  • Franco Flandoli
  • (Scuola Normale Superiore, Pisa)
  • Mixing and dissipation properties of transport noise
  • Abstract:
  • This talk is based on recent works with Dejun Luo and Lucio Galeati devoted to the investigation of a suitable scaling limit of several different PDE models subject to transport noise, when the noise is extremized in a suitable limit sense. Among the consequences there are certain forms of mixing, enhanced dissipation, delayed blow-up due to noise; these results hold for several classes of equations including Euler and Navier-Stokes equations, Keller-Siegel and reaction diffusion equations; and also rigorous results on eddy viscosity and eddy dissipation in turbulent fluids have been proved. Along with arguments of stochastic model reduction, developed with Umberto Pappalettera, a picture arises of the potential effects of small scale fluctuations on large scale properties of turbulent fluids.
  • 27.04.21   09:00

  • Mythily Ramaswamy
  • (TIFR Centre for Applicable Mathematics, Bangalore)
  • Local stabilization of time periodic flows
  • Abstract:
  • Fluid flows have been studied for a long  time, with a view to understand better the models like channel flow, blood flow, air flow in the lungs etc. Here we focus on  time periodic fluid flow models. Local stabilization here concerns the decay of the perturbation in the flow near a periodic trajectory. The main motivating example is the incompressible Navier-Stokes system. I will discuss the general framework to study periodic solutions and then indicate some results in this direction.
  • 20.04.21   09:00

  • Francesco Fanelli
  • (University of Lyon)
  • Statistical solutions to the barotropic Navier-Stokes equations
  • Abstract:
  • In this talk we are concerned with the notion of statistical solutions to some models of fluid mechanics. We focus on the barotropic Navier-Stokes equations, supplemented with non-homogeneous boundary data.
    In the first part of the talk, we study dynamical properties of statistical solutions. Our approach, different from the classical one of Foia?-Prodi and Vishik-Fursikov for the incompressible system, is based on a semiflow selection procedure. This allows us to define statistical solution as the push-forward measure of the initial probability distribution on the space of data of the Navier-Stokes system. We then investigate questions like existence and stability of
    statistical solutions.
    In the second part of the talk, we focus on the special class of stationary statistical solutions. In particular, we explore their role in the investigation of the validity of the so-called ergodic hypothesis in the context of the barotropic Navier-Stokes equations.

    This talk is based on joint works with Eduard Feireisl (Czech Academy of Sciences) and Martina Hofmanová (Universität Bielefeld).
  • 13.04.21   09:00

  • Martina Hofmanová
  • (University of Bielefeld)
  • Non-uniqueness in law of stochastic 3D Navier-Stokes equations
  • Abstract:
  • We consider the stochastic Navier-Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on two iconic examples of a stochastic perturbation: either an additive or a linear multiplicative noise driven by a Wiener process. In both cases, we develop a stochastic counterpart of the convex integration method  introduced recently by Buckmaster and Vicol. This permits to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,infty). Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non-uniqueness in law holds on an arbitrary time interval [0,T], T>0.
  • 06.04.21   09:00

  • Antonín Novotný
  • (University of Toulon)
  • Compressible fluids with nonhomogeneous boundary data II
  • Abstract:
  • We shall discuss several problems in the mathematical analysis of  viscous compressible fluids under the action of non zero inflow-outflow boundary conditions.

    The lecture will be delivered on Zoom (only):
    Meeting ID: 944 2924 8803
    Passcode: 697514

    NEW: The record of the lecture
  • 30.03.21   09:00

  • Antonín Novotný
  • (University of Toulon)
  • Compressible fluids with nonhomogeneous boundary data I
  • Abstract:
  • We shall discuss several problems in the mathematical analysis of  viscous compressible fluids under the action of non zero inflow-outflow boundary conditions.

    The lecture will be delivered on Zoom (only):
    Meeting ID: 944 2924 8803
    Passcode: 697514

    NEW: The record of the lecture
  • 23.03.21   08:30

  • Michele Coti Zelati
  • (Imperial College London)
  • Stationary Euler flows near the Kolmogorov and Poiseuille flows
  • Abstract:
  • We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow. This is in contrast with both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel, for which we can show that the only stationary states near them must be shears. This has surprising consequences in the context of inviscid damping in 2D Euler and enhanced dissipation in Navier-Stokes.

    NEW: The slides of the lecture
  • 16.03.21   09:00

  • Václav Mácha
  • (Institute of Mathematics, CAS)
  • Local-in-time existence of strong solutions to a class of compressible non-Newtonian Navier-Stokes equations
  • Abstract:
  • We show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equation for arbitrarily large initial data. The goal is reached by $L^p$-theory for linearized equations which are obtained with help of the Weis multiplier theorem. This work was done in collaboration with M. Kalousek and Š. Nečasová.
  • 09.03.21   09:00

  • David Lannes
  • (Université de Bordeaux)
  • Some problems arising in wave-structure interactions
  • Abstract:
  • There are different formulations of the water waves problem. One of them is to formulate it as a system of equations coupling two quantities, e.g. the free surface elevation $zeta$ and the horizontal discharge $Q$. Actually, one can understand the water waves problem as a system on three quantities, $zeta$, $Q$ and the surface pressure $P_s$ under the constraint that $P_s$ is constant (and therefore disappears from the equations).
    When we consider in addition a floating body then, under the body, we still have a system of equations on the same three quantities, but this time the constraint is not on the pressure but on the surface of the water, that must coincide with the bottom of the floating object.
    Wave-structure interactions can be understood as the coupling of these two different constrained problems. We shall briefly analyse this coupling and show among other things how it dictates the evolution of the contact line between the surface of the water and the surface of the floating body, and how to transform it into transmission problems that raise many mathematical issues such as fully nonlineary hyperbolic initial boundary value problems, dispersive boundary layers, initial boundary value problems for nonlocal equations, etc.
  • 02.03.21   09:00

  • Piotr B. Mucha
  • (University of Warsaw)
  • Flows initiated by ripped densities
  • Abstract:
  • We address the question: Are solutions to the equations of viscous flows that are initiated by a density function given by a characteristic function of a set regular and unique?  The positive answer is possible for the compressible Navier-Stokes equations if the bulk/volume viscosity is large. The limit case of the homogeneous incompressible NSEs will be discussed too.

    The talk will be based on results with Raphael Danchin:
    RD, PBM: Compressible NSEs with ripped density, arXiv;
    RD, PBM: The incompressible NSEs in vacuum, CPAM2019.
  • 05.01.21   09:00

  • Raphaël Danchin
  • (Université Paris-Est Créteil)
  • A class of global relatively smooth solutions for the Euler-Poisson system
  • Abstract:
  • In this joint work with X. Blanc, B. Ducomet and Š. Nečasová (to appear in JHDE), we construct a class of global solutions to the Cauchy problem for the isentropic Euler equations coupled with the Poisson equation, in the whole space. The initial density is assumed to decay to 0 at infinity and the initial velocity is close to some reference velocity with Jacobian having positive spectrum bounded away from 0. By a suitable adaptation of Grassin-Serre’s work on the `pure’ compressible Euler equations, we obtain a global smooth solution  the large time behavior of which may be described in terms of some solution of the multi-dimensional Burgers equation. The stability of some special spherically symmetric stationary solution is also discussed.
  • 15.12.20   09:00

  • Tong Tang
  • (Hohai University, Nanjing)
  • Global existence of weak solutions to the quantum Navier-Stokes equations
  • Abstract:
  • In this talk, we proved the global existence of weak solutions to the quantum Navier-Stokes equations with non-monotone pressure. Motivated by the work of Antonell-Spirito (2017, Arch. Ration. Mech. Anal., 1161-1199) and Ducomet-Necasova-Vasseur (2010, Z. Angew. Math. Phys., 479-491), we construct the suitable approximate system and obtain the corresponding compactness by B-D entropy estimate and Mellet-Vasseur inequality.
  • 08.12.20   09:00

  • Martin Kalousek
  • (Institute of Mathematics, CAS)
  • Global existence of weak solutions for a magnetic fluid model
  • Abstract:
  • The talk is devoted to the presentation of recent results that concern the global in time existence of weak solutions of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier-Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn-Hilliard equations. Presented results are based on the joint work with S. Mitra and A. Schlömerkemper.
  • 01.12.20   09:00

  • Marco Bravin
  • (Basque Center for Applied Mathematics)
  • Interaction of a small rigid body with fluids
  • Abstract:
  • In this talk I will present a recent result in collaboration with Prof Necasova, where we study the interaction between a small rigid body and a compressible viscous fluid modeled by the compressible Navier-Stokes equations.

    In particular I will recall the previous results where the fluids were supposedly incompressible and then I will focus my attention on the improved pressure estimates that are the main novelty in our result. In contrast with the incompressible case the pressure estimates depend on a lower bound of the mass and the inertia matrix of the object as its size tends to zero.
  • 24.11.20   09:00

  • Tomasz Piasecki
  • (University of Warsaw)
  • A maximal regularity approach to compressible mixtures
  • Abstract:
  • I will present recent results obtained in collaboration with Yoshihiro Shibata and Ewelina Zatorska. We investigate the well posedness of a system describing flow of a mixture of compressible constituents. The system in composed of Navier-Stokes equations coupled with equations describing balance of fractional masses. A crucial property is that the system is non-symmetric and only degenerate parabolic.

    However, it reveals a structure which allows to transform it to a symmetric parabolic problem using appropriate change of unknowns. In order to treat the transformed problem we write it in Lagrangian coordinates and linearize. For the related linear problem we show a Lp-Lq maximal regularity estimate applying the theory of R-bounded solution operators. This estimate allows to show local existence and uniqueness. Next, assuming additionally boundedness of the domain we extend the maximal regularity estimate and show exponential decay  property for the linear problem. This allows us to show global well-posedness of the original problem for small data.
  • 03.11.20   09:00

  • Gianmarco Sperone
  • (Charles University)
  • Explicit bounds for the generation of a lift force exerted by steady-state Navier-Stokes flows over a fixed obstacle
  • Abstract:
  • We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian.
    The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalŕ (both at Politecnico di Milano).
  • 20.10.20   09:00

  • Ralph Chill
  • (TU Dresden)
  • Degenerate gradient systems: the bidomain problem and Dirichlet-to-Neumann operators
  • Abstract:
  • We identify a common structure behind several parabolic PDE models involving Dirichlet-to-Neumann operators, the bidomain operator, and some more, and we show that they actually have a gradient structure, possibly up to lower order perturbations. This has consequences for wellposedness of these PDEs, regularity of solutions and their asymptotic behaviour.
  • 13.10.20   09:00

  • Marija Galić
  • (University of Zagreb)
  • Existence of a weak solution to a 3d nonlinear, moving boundary FSI problem
  • Abstract:
  • See the attachment.
  • 06.10.20   09:00

  • Paolo Maria Mariano
  • (University of Florence)
  • Varifold-based variational description of crack nucleation: results and perspectives
  • Abstract:
  • In crack nucleation and growth processes, the crack margins may be in contact although not linked by material bonds. In variational descriptions of fractures in elastic-brittle bodies -descriptions all based on energy minimization and De Giorgi's view on minimizing movements - the circumstance can be hardly described just by identifying cracks with the jump set of special bounded variation functions. On the other hand, if we consider both crack paths and deformations as distinct entities - although linked because the deformation jump set is at least included in the crack path - we face the basic problem of controlling minimizing sequences of crack surfaces in three-dimensional environment. The problem can be eluded if we can take - roughly speaking - just sequences of surfaces with bounded curvature, although we should think of the notion of curvature we use because crack surfaces could be extremely rough. To this aim, we can consider crack paths as rectifiable sets supporting vector-valued measures, which admit a generalized notion of curvature: they are the so-called "curvature varifolds". Their introduction suggest a form of the energy for elastic-brittle bodies modifying standard Griffith's one, but  maintaining its polyconvex structure with respect to the deformation gradient. For such an energy it is possible to find existence theorems for minimizers, which are couples deformation/varifold. In this view, deformations can be both weak diffeomorphisms and SBV-based weak diffeomorphisms. In this way, we can describe from a variational view the nucleation of fractures without introducing additional failure criteria. In my talk, I'll describe such a setting with pertinent results, offering some possible research perspectives.
  • 25.02.20   10:15

  • Minsuk Yang
  • (Yonsei University)
  • New Regularity Criteria for Weak Solutions to the MHD Equations in Terms of an Associated Pressure
  • Abstract:
  • We prove that a suitable weak solutions of the three-dimensional MHD equations are smooth if the negative part of the pressure is suitably controlled.
  •                    09:00

  • Tongkeun Chang
  • (Yonsei University)
  • On Caccioppoli's inequalities of Stokes equations and Navier-Stokes equations near boundary
  • Abstract:
  • We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's  inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary.
  • 18.02.20   09:00

  • Srđan Trifunović
  • (Shanghai Jiao Tong University)
  • On the interaction of compressible fluids and nonlinear thermoelastic plates
  • Abstract:
  • in the attachment
  • 28.01.20   09:00

  • Tuhin Ghosh
  • (Hong Kong University of Science and Technology)
  • The Fractional Calderon problem
  • Abstract:
  • We will be discussing the fractional Calderon problem, where one tries to determine an unknown potential in a fractional Schrodinger equation from the exterior measurements of solutions.
  • 17.12.19   10:15

  • Richard Andrášik
  • (Palacký University Olomouc)
  • Compressible nonlinearly viscous fluids: Asymptotic analysis in a 3D curved domain
  • Abstract:
  • Governing equations representing mathematical description of continuum mechanics have often three spatial dimensions and one temporal dimension. However, their analytical solution is usually unattainable, and numerical approximation of the solution unduly complicated and computationally demanding. Thus, these models are frequently simplified in various ways. One option of a simplification is a reduction of the number of spatial dimensions. Nonsteady Navier-Stokes equations for compressible nonlinearly viscous fluids in a three-dimensional domain were considered. These equations need a simplification, when possible, to be effectively solved. Therefore, a dimension reduction was performed for this type of a model. Dynamics of a compressible fluid in thin domains was studied. The current framework was extended by dealing with nonsteady Navier-Stokes equations for compressible nonlinearly viscous fluids in a deformed three-dimensional domain where only two dimensions are dominant. The deformation of a domain introduced new difficulties in the asymptotic analysis, because it affects the limit equations in a non-trivial way. However, these challenges were addressed, and the two-dimensional model was rigorously derived.
  •                    09:00

  • Marília Pires
  • (Institute of Mathematics, CAS)
  • Influence of rheological parameters on generalized Oldroyd-B fluid flow through curved pipes
  • Abstract:
  • Flows in curved pipes are very challenging and considerably more complex than flows in straight pipes. Due to fluid inertia, a secondary motion appears in addition to the primary axial flow. It is induced by an imbalance between the cross stream pressure gradient and the centrifugal force and consists of a pair of counter-rotating vortices, which appear even for the most mildly curved pipe.

    Parallel to the pipe curvature ratio, the rheological parameters of the fluid have a considerable influence on the flow behavior. In this work, numerical simulations obtained by finite elements method, involving steady, incompressible, creeping and inertial flows of the generalized Oldroyd-B fluid  through curved pipes are presented. The behavior of the solutions is discussed with respect to different rheologic and geometric flow parameters.
  • 10.12.19   09:00

  • Marcel Braukhoff
  • (Vienna University of Technology)
  • Chemotaxis-consumption model and the importance of the boundary conditions
  • Abstract:
  • In the talk we discuss the behavior of the concentration of some bacteria swimming in water (for example of the species Bacillus subtilis), whose otherwise random motion is partially directed towards higher concentrations of a signaling substance (oxygen) they consume. After a transition phase, the system can be described using a chemotaxis-consumption model on a bounded domain. Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed.

    Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, this talks considers no-flux boundary conditions for the bacteria density and the physically meaningful Robin boundary conditions for the signaling substance and Dirichlet boundary conditions for the flow.

    In the talk, we study the existence of a global (weak) solution. Moreover, we discuss how to show that there exists a unique stationary solution for any
    given mass assuming that the flow vanishes. This solution is non-constant. In the radial symmetric case, the densities are strictly convex.
  • 26.11.19   10:15

  • Lisa Beck
  • (University of Augsburg)
  • Lipschitz bounds and non-uniform ellipticity
  • Abstract:
  • In this talk we consider a large class of non-uniformly elliptic variational problems and discuss optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the data. The analysis covers the main model cases of variational integrals of anisotropic growth, but also of fast growth of exponential type investigated in recent years. The regularity criteria are established by potential theoretic arguments, involve natural limiting function spaces on the data, and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation.
    The results presented in this talk are part of a joined project with Giuseppe Mingione (Parma).
  •                    09:00

  • Stefan Krömer
  • (Institute of Information Theory and Automation, CAS)
  • Injective nonlinear elasticity via penalty terms: analysis and numerics
  • Abstract:
  • I will present some new ideas for static nonlinear elasticity with a global injectivity constraint preventing self-interpenetration of the elastic body. Our main focus are penalization terms replacing this injectivity constraint, the Ciarlet-Nečas condition. For models of non-simple materials which include a term with higher order derivatives, the penalized model is shown to converge to the constrained original model. Among other things, the penalization can be chosen in such a way that self-interpenetration is prevented even at finite value of the penalization parameter, and not just in the limit. Our penalty method also provides a working numerical scheme with provable convergence along a subsequence.

    This is joint work with Jan Valdman (UTIA CAS).
  • 19.11.19   09:00

  • Marco Bravin
  • (University of Bordeaux)
  • On the asymptotic limit of a shrinking source and sink in a 2D bounded domain
  • Abstract:
  • In this talk I will present a recent result on the study of the asymptotic limit of a shrinking source and sink in a perfect two dimensional fluid. The system consists of an Euler type system in a bounded domain with two holes and non-homogeneous boundary conditions are prescribed on the boundary. These conditions lead to the creation of a point source and a vortex point in the limit. Similar type of systems have been already study by Chemetov and Starovoitov in [1], where a different approximation approach was considered.

    [1] Chemetov, N. V., Starovoitov, V. N. (2002). On a Motion of a Perfect Fluid in a Domain with Sources and Sinks. Journal of Mathematical Fluid Mechanics, 4(2), 128-144.
  • 22.10.19   09:00

  • Václav Mácha
  • (Institute of Mathematics, CAS)
  • On a body with a cavity filled with compressible fluid
  • Abstract:
  • We discuss the dynamics of a hollow body filled with compressible fluid. The main aim of our effort is to investigate the long time behaviour of the whole system. At first, we show the existence of weak and strong solutions and we show the weak-strong uniqueness principle. We investigate the steady case which helps to deduce the possible long-time limits. The semigroup approach then allows to rigorously examine the long time behaviour. At last, the aforementioned method is used also to a system consisting of a hollow pendulum filled with a compressible fluid. The presented talk is based on results obtained in collaboration with Š. Nečasová and G. P. Galdi.
  • 15.10.19   09:00

  • Nicola Zamponi
  • (Charles University)
  • A non-local diffusion equation
  • Abstract:
  • We consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator in 2 space dimensions. Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality, a higher integrability estimate of the approximate solution and a generalization of the well-known Div-Curl Lemma.
  • 08.10.19   09:00

  • Amrita Ghosh
  • (Institute of Mathematics, Czech Academy of Sciences)
  • L^p-Strong solution to fluid-rigid body interaction system with Navier slip boundary condition
  • Abstract:
  • I will discuss the existence of a strong solution of a coupled fluid and rigid body system and the corresponding L^p-theory. Precisely, I will consider a 3D viscous, incompressible non-Newtonian fluid, containing a 3D rigid body, coupled with (non-linear) slip boundary condition at the interface and show the well-posedness of this system.
  • 01.10.19   09:00

  • Lukáš Kotrla 
  • (University of West Bohemia, Pilsen)
  • Strong maximum principle for problem involving $p$-Laplace operator
  • Abstract:
  • We consider continuous nonnegative solutions to a doubly nonlinear parabolic problem with the $p$-Laplacian with zero Dirichlet boundary conditions. For simplicity we assume that both the initial data and the reaction function are continuous and nonnegative and the reaction function does not depend on $u$. We show that for $1<p<2$ the speed of propagation is infinite in the sense that for any fixed time the solution is either everywhere positive or identically zero. In particular, if the initial data are nonzero at at least one point, then for small positive time the solution is positive in the whole domain, i.e., the strong maximum principle holds. We will also apply maximum and comparison principles to problems from turbulent filtration of natural gas in porous rock and groundwater filtration in gravel. In particular, we will focus on a model of turbulent filtration of natural gas in a porous rock due to Leibenson. This is a joint work with P. Girg and P. Takac.
  • 21.05.19   09:00

  • Chérif Amrouche
  • (University of Pau and Pays de l'Adour)
  • Harmonic and Biharmonic Problems in Lipschitz and C^{1,1} Domains
  • Abstract:
  • See the attachment.
  • 14.05.19   09:00

  • Martin Kružík
  • (Institute of Information Theory and Automation, Czech Academy of Sciences)
  • On the passage from nonlinear to linearized viscoelasticity
  • Abstract:
  • We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin‘s-Voigt‘s rheology where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. Main tools used are rigidity estimates and gradient flows in metric spaces. This is a joint work with M. Friedrich (Munster).
  • 30.04.19   09:00

  • Vladimir Bobkov
  • (University of West Bohemia)
  • On Payne's nodal set conjecture for the p-Laplacian
  • Abstract:
  • The Payne conjecture asserts that the nodal set of any second eigenfunction of the zero Dirichlet Laplacian intersects the boundary of the domain. We prove this conjecture for the p-Laplacian assuming that the domain is Steiner symmetric. (In particular, the domain can be a ball.) The talk is based on the joint work with S. Kolonitskii.
  • 16.04.19   09:00

  • Nicola Zamponi
  • (Charles University)
  • Analysis of a degenerate and singular volume-filling cross-diffusion system modeling biofilm growth
  • Abstract:
  • We analyze the mathematical properties of a multi-species bio?lm cross-di?usion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of volume-?lling type cross-di?usion systems which exhibit a porous medium-type degeneracy when the total biomass vanishes as well as a superdi?usion-type singularity when the biomass reaches its maximum cell capacity. The equations also admit a very interesting non-standard entropy structure. We prove the existence of global-in-time weak solutions, study the asymptotic behavior and the uniqueness of the solutions, and complement the analysis by numerical simulations that illustrate the theoretically obtained results.
  • 09.04.19   09:00

  • Petr Pelech
  • (Charles University)
  • Getting familiar with the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC)
  • Abstract:
  • One common feature of new emerging technologies is the fusion of the very small (nano) scale and the large scale engineering. The classical enviroment provided by single scale theories, as for instance by the classical hydrodynamics, is not anymore satisfactory. It is the main goal of GENERIC to provide a suitable framework for developing and formulating new thermodynamic models [1]. As an inevitable consequence, the mathematical nature of these new models is different. For instance, the governing equations cannot be written as conservation laws and hence finding a new suitable mathematical structure is necessary. A possible solution seems to be given by the so-called Symmetric Hyperbolic Thermodynamical Consistent (SHTC) equations [2], for which local well-posedness is known [3-6]. There are also numerical computations based on the discontinuous Galerkin method [7], however, a rigorous mathematical analysis of the global-in-time existence has still not been developed.

    In this introductory talk I will try to explain some of the GENERIC's fundamental notions and important principles on very simple examples. I will also show on the well established models (compressible Euler or Navier-Stokes, viscoelastic Maxwell) how to manipulate equations in the GENERIC framework.

    [1] Pavelka, M., Klika, V. & Grmela, M. (2018). Multiscale Thermo-Dynamics. Introduction to GENERIC. Berlin, Boston: De Gruyter. Retrieved 28 Mar. 2019, from https://www.degruyter.com/view/product/254928
    [2] Peshkov, I., Pavelka, M., Romenski, E., Grmela, M.: Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Continuum Mechanics and Thermodynamics 30(6), 1343--1378 (2018). DOI 10.1007/s00161-018-0621-2. URL https://doi.org/10.1007/s00161-018-0621-2
    [3] Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Archive for Rational Mechanics and Analysis 58(3), 181–205 (1975). DOI 10.1007/BF00280740. URL http://link.springer.com/10.1007/BF00280740
    [4] Benzoni-Gavage, S., Serre, D.: Multidimensional Hyperbolic Partial Differential Equations. Oxford University Press, Oxford (2007)
    [5] Muller, I., Ruggeri, T.: Rational Extended Thermodynamics, vol. 16. Springer (1998)
    [6] Romenski, E., Drikakis, D., Toro, E.: Conservative Models and Numerical Methods for Compressible Two-Phase Flow. Journal of Scientific Computing 42(1), 68–95 (2010)
    [7] Dumbser, M.; Fambri, F.; Tavelli, M.; Bader, M.; Weinzierl, T. Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine. Axioms 7(3):63, (2018). DOI 10.3390/axioms7030063
  • 02.04.19   09:00

  • Yang Li
  • (Institute of Mathematics, CAS)
  • Some results on compressible magnetohydrodynamic system with large initial data
  • Abstract:
  • The time evolution of electrically conducting compressible flows under the mutual interactions with the magnetic field is described by the system of magnetohydrodynamics (MHD). In this talk, we focus on the existence of global-in-time weak solutions with large initial data. Firstly, for a simplified 2D MHD model of viscous non-resistive flows, we prove the existence of global-in-time weak solutions by invoking the idea from compressible two-fluid model. Secondly, for the general 3D inviscid resistive MHD system, we prove the existence of infinitely many global-in-time weak solutions for any smooth initial data. To do this, we appeal to the method of convex integration developed by De Lellis and Szekelyhidi and adapted to the compressible flows by Chiodaroli, Feireisl and Kreml.
    The results are based on the joint works with Eduard Feireisl and Yongzhong Sun.
  • 26.03.19   09:00

  • Aneta Wroblewska-Kamińska
  • (Institute of Mathematics, Polish Academy of Sciences)
  • The incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids
  • Abstract:
  • We explore the behaviour of global-in-time weak solutions to a class of bead-spring chain models, with finitely extensible nonlinear elastic (FENE) spring potentials, for dilute polymeric fluids. In the models under consideration the solvent is assumed to be a compressible, isentropic, viscous, isothermal Newtonian fluid, confined to a bounded open domain in R^3, and the velocity field is assumed to satisfy a complete slip boundary condition. We show that for ill-prepared initial data, as the Mach number tends to zero, the system is driven to its incompressible counterpart.
    The result is a joint work with Endre Süli.
  • 19.03.19   09:00

  • Ondřej Kreml
  • (Institute of Mathematics, CAS)
  • Wild solutions to isentropic Euler equations starting from smooth initial data
  • Abstract:
  • We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of smooth initial data which admit infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant T > 0. In order to continue the solution after the formation of the discontinuity, we apply the theory developed by De Lellis and Szekelyhidi and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution.
    This is a joint work with Elisabetta Chiodaroli, Václav Mácha and Sebastian Schwarzacher.
  • 12.03.19   09:00

  • Giselle Monteiro
  • (Institute of Mathematics, CAS)
  • On Preisach operators and piezoelectricity modeling
  • Abstract:
  • Preisach operators are rate independent hysteresis operators capable of reproducing minor loops, therefore they are well fitted to measurements of smart materials. Benefiting from this observation, some authors have proposed models for piezoelectricity assuming that all hysteresis effects are due to one single Preisach operator. More accurate models though have to account thermal effects. To address this problem, we introduce a notion of parameter-dependent Preisach operator and investigate some properties of its inverse.
    This is a joint work with P. Krejci.
  • 11.12.18   09:00

  • Martin Fencl
  • (University of West Bohemia, Pilsen)
  • Unilateral sources of an activator in reaction-diffusion systems describing Turing’s patterns
  • Abstract:
  • See the attachment.
  • 04.12.18   10:00

  • Ondřej Kreml
  • (Institute of Mathematics, CAS)
  • Wild solutions for isentropic Euler equations starting from smooth initial data
  • Abstract:
  • In a series of papers starting with the groundbreaking work of De Lellis and Székelyhidi several authors have shown that there might exist infinitely many bounded weak solutions to the isentropic Euler equations satisfying the energy inequality and starting from certain class of initial data. Concerning smoothness, the best result is due to Chiodaroli, De Lellis and Kreml, where the existence of these wild solutions was shown for Lipschitz initial data. In this talk we present the same result for smooth initial data. The proof is based on a nontrivial generalization of the previous theorem, in particular on a notion of generalized fan subsolution.
    This is a joint work with E. Chiodaroli, V. Mácha and S. Schwarzacher.
  • 27.11.18   09:00

  • Xavier Blanc
  • (Université Paris-Diderot)
  • Homogenization in the presence of defects
  • Abstract:
  • We will present some results on homogenization for linear elliptic equation. The geometry will be assumed to be either a perturbation of a periodic background, or an interface between two periodic media. In both cases, we study the homogenization problem, prove existence of a corrector, and use to build a two-scale expansion of the solution. We prove convergence estimates of this two-scale expansion.
    These are a joint works with C. Le Bris (Ecole des Ponts, Paris), P.-L. Lions (Collčge de France, Paris) and M. Josien (Ecole des Ponts, Paris).
  • 20.11.18   09:00

  • Bangwei She
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Convergence of a finite volume scheme for the compressible Navier-Stokes system
  • Abstract:
  • We study the convergence of a finite volume scheme for the compressible (barotropic) Navier--Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system.  Then by the weak-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.
    This is a joint work with E. Feireisl, M. Lukacova and  H. Mizerova.
  • 13.11.18   09:00

  • Václav Mácha
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Body with a Cavity Filled with a Compressible Fluid
  • Abstract:
  • We  study  the  dynamics  of  a  system  composed  by a rigid body containing a visous compressible fluid. The  emphasis  is  laid  upon  the  analysis  of  the  long time behavior of the whole system.  We show that for small initial data the whole system tends to a permanent rotation similarly as in the incompressible case. On the other hand, we highlight some problems coming from compressibility which do not allow to prove the same for solutions emanating from arbitrary initial data.
    The work was done in collaboration with G. P. Galdi and S. Necasova
  • 30.10.18   09:00

  • Ansgar Jüngel
  • (TU Wien)
  • Analysis of diffusive population systems for multiple species
  • Abstract:
  • The dynamics of multi-species populations can be described by random walks on a lattice which leads in the diffusive limit to nonlinear reaction-cross-diffusion systems. A special model was suggested by Shigesada, Kawasaki, and Teramoto in 1979. The diffusion matrix of these cross-diffusion systems is typically neither symmetric nor positive definite, which complicates the analysis. The idea is to reveal a so-called entropy structure (which is a special Lyapunov functional) allowing for gradient estimates. In this talk, we review recent results on population cross-diffusion models, including the local and global existence analysis, uniqueness of weak solutions, and their large-time asymptotics.
  • 23.10.18   09:00

  • Tong Tang
  • (Institute of Mathematics, Czech Academy of Sciences)
  • On a singular limit for the stratified compressible Euler system
  • Abstract:
  • We consider a singular limit for the compressible Euler system in the low Mach number regime driven by a large external force. We show that any dissipative measure-valued solution approaches a solution of the lake equation in the asymptotic regime of low Mach and Froude numbers. The result holds for the ill-prepared initial data creating rapidly oscillating acoustic waves. We use dispersive estimates of Strichartz type to eliminate the effect of the acoustic waves in the asymptotic limit.
  • 16.10.18   09:00

  • Tomasz Piasecki
  • (University of Warsaw)
  • Strong solutions to the steady compressible Navier-Stokes equations with inflow boundary conditions
  • Abstract:
  • We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition in a vicinity of given laminar solutions under the assumption that the pressure is a linear function of the density. In particular, we do not require any information on the gradient of the density or second gradient of the velocity. Our result holds provided certain condition on the shape of the boundary around the points where characteristics of the continuity equation are tangent to the boundary, which holds in particular for piecewise analytical boundaries.
  • 02.10.18   09:00

  • Jiří Neustupa
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Spectral instability of a steady flow of an incompressible viscous fluid past a rotating obstacle
  • Abstract:
  • We show that a steady solution U to the system of equations of motion of an incompressible Newtonian fluid past a rotating body is unstable if an associated linear operator L has at least one eigenvalue in the right half-plane in C. Our theorem does not directly follow from a series of preceding results on instability, mainly because the associated nonlinear operator is not bounded in the same space in which the instability is studied. As an important auxiliary result, we also show that the uniform growth bound of the C_0 semigroup e^{Lt} is equal to the spectral bound of operator L.
  • 15.05.18   09:00

  • Lucio Boccardo
  • (Sapienza University of Rome)
  • Regularizing effect of the lower order terms in some nonlinear Dirichlet problems
  • Abstract:
  • See the attachment.
  • 11.05.18   10:00

  • Zdeněk Skalák
  • (Czech Technical University)
  • Regularity criteria for the Navier-Stokes equations in terms of the velocity gradient
  • Abstract:
  • See the attachment.
  • 24.04.18   09:00

  • Petr Pelech
  • (Institute of Information Theory and Automation, Czech Academy of Sciences)
  • Gradient Polyconvexity in the Framework of Rate-Independent Processes
  • Abstract:
  • The talk treats mathematical aspects of evolutionary material models for shape-memory alloys at finite-strains. The difficulty of related mathematical analysis consists in the non-linear and non-convex dependence of the energy on the deformation gradient. One possible way, how maintain the analysis tractable, is to suppose that the energy depends also on the second deformation gradient and is convex in it. We relax this assumption by using the recently proposed concept of gradient-polyconvexity. Namely, we consider energies which are convex only in gradients of non-linear minors (i.e. cofactor and determinant in three dimension) of the deformation gradient. As a result, the whole second deformation gradient needs not to be integrable. Yet, at the same time, the obtained compactness is sufficient and, moreover, additional physically desirable properties(e.g. local invertibility) can be shown. We extend the previous result for hyperelastic materials by incorporating a rate-independent dissipation to our model and by proving existence of an energetic solution to it. It is a joint work with Martin Kružík (Institute of Information Theory and Automation of the Czech Academy of Sciences) and Anja Schloemerkemper (University of Wuerzburg).
  • 10.04.18   09:00

  • Miroslav Bulíček
  • (Charles University)
  • Nonlinear ellitpic and parabolic equations beyond the natural duality pairing
  • Abstract:
  • Many real-world problems are described by nonlinear partial differential equations. A promiment example of such equations is nonlinear (quasilinear) elliptic system with given right hand side in divergence form div f data. In case data are good enough (i.e., belong to L^2), one can solve such a problem by using the monotone operator therory, however in case data are worse no existence theory was available except the case when the operator is linear, e.g. the Laplace operator. For this particular case one can however establish the existence of a solution whose gradient belongs to L^q whenever f belongs to L^q as well. From this point of view it would be nice to have such a theory also for general operators. However, it cannot be the case as indicated by many counterexamples. Nevertheless, we show that such a theory can be built for operators having asymptotically the radial structrure, which is a natural class of operators in the theory of PDE. As a by product we develop new theoretical tools as e.g., weighted estimates for the linear problems and the new compensated compactness method represented by the div-curl-biting-weighted lemma.
  • 27.03.18   09:00

  • Simon Markfelder
  • (Julius-Maximilians-Universität Würzburg)
  • Non-uniqueness of entropy solutions to the 2-d Riemann problem for the Euler equations
  • Abstract:
  • In this talk we consider the compressible (full) Euler equations in two space dimensions together with Riemann initial data. The issue of the talk is the question on uniqueness of weak entropy solutions to this problem. This issue has been studied for the isentropic Euler equations by E. Chiodaroli, C. De Lellis and O. Kreml (among others) and the aim is now to extend the results to full (i.e. non-isentropic) Euler. We consider a special class of Riemann data, namely those for which the 1-d self-similar solution consists of two shocks and possibly a contact discontinuity. We show that for this class there exist infinitely many weak entropy solutions, which are generated by convex integration. This is joint work with H. Al Baba, C. Klingenberg, O. Kreml and V. Mácha.
  • 20.03.18   09:00

  • Jan Březina
  • (Tokyo Institute of Technology)
  • Measure-valued solutions and Navier-Stokes-Fourier system
  • Abstract:
  • Encouraged by the ideas and results obtained when studying measure-valued solutions for the Complete Euler system we introduce measure-valued solutions to the Navier–Stokes–Fourier system and show weak-strong uniqueness. Namely, we identify a large class of objects that we call dissipative measure–valued (DMV) solutions, in which the strong solutions are stable. That is, a (DMV) solution coincides with the strong solution emanating from the same initial data as  long as the latter exists.
  • 13.03.18   09:00

  • Erika Maringová
  • (Charles University)
  • Up to the boundary Lipschitz regularity for variational problems
  • Abstract:
  • We prove the existence of a regular solution to a wide class of convex, variational integrals. Via technique of construction of the barriers we show that the  solution is Lipschitz up to the boundary. For the linear growth case, we identify the necessary and sufficient condition to existence of solution; in the case of superlinear growth, we provide the sufficient one. The result is not restricted to any geometrical assumption on the domain, only its regularity plays the role.
  • 06.03.18   09:00

  • Gabrielle Brüll
  • (Norwegian University of Science and Technology)
  • Weak solutions to a two-phase thin film equation with insoluble surfactant
  • Abstract:
  • We discuss a model describing the spreading of an insoluble surfactant on the upper surface of a viscous complete wetting two-phase thin film. Considering capillary effects as the only driving force, the system of evolution equations consists of two strongly coupled degenerated equations of fourth order describing the film heights of the fluids, which are additionally coupled to a second-order transport equation for the surfactant concentration. Owing to the degeneracy, it is in general not clear whether one can prove the existence of global solutions in a classical sense, which motivates the study of weak solutions. The proof of existence of nonnegative global weak solutions is based on a priori energy estimates and compactness arguments.
  • 27.02.18   09:00

  • Dalibor Pražák
  • (Charles University)
  • Regularity and uniqueness for a critical Ladyzhenskaya fluid
  • Abstract:
  • We consider an incompressible p-law type fluid in a 3D bounded domain. Employing iterative estimate in Nikolskii spaces and reverse Hölder inequality, we establish higher time regularity and uniqueness of weak solution provided the data are more regular.
    This is a joint work with M. Bulíček and P. Kaplický.
  • 20.02.18   09:00

  • Petr Kučera
  • (Czech Technical University)
  • Some properties of the strong solution of the Navier-Stokes equations
  • Abstract:
  • TBA
  • 09.01.18   09:00

  • Matteo Caggio
  • (Institute of Mathematics, CAS)
  • Navier-Stokes equations and related problems in mathematics and physics
  • Abstract:
  • We will deal with Navier-Stokes equations for compressible and incompressible fluids and we will discuss some related problems in mathematics and physics. In the mathematical context, we will discuss singular limits problems for compressible fluids, fluids subjected to dimension reduction and conditional regularity for incompressible Navier-Stokes equations in the whole space. Some working progress in the context of Rayleigh-Benard problem for micropolar fluids, weak-strong uniqueness for fluid-structure interaction and thin-layer approximation will be also presented. In the physical context, we will discuss problems related to the turbulent boundary layer. In particular, the Camassa-Holm equations for a flat-plate boundary layer and the similarity theory for a stationary atmospheric surface-layer over horizontally homogeneous terrain.
  • 12.12.17   09:00

  • Ondřej Kreml
  • (Institute of Mathematics, CAS)
  • Recent progress in the analysis of the Riemann problem for the 2D compressible Euler equations
  • Abstract:
  • The Riemann problem presents a basic tool in the analysis of hyperbolic conservation laws in 1D. Recently, it has been used also to demonstrate the ill behaviour of incompressible and compressible Euler systems in multiple dimensions, where the convex integration theory of De Lellis and Székelyhidi can be applied. Apart from being just a tool, the Riemann problem for compressible isentropic Euler system itself provides an interesting problem. It can be shown that for some Riemann data the problem admits a unique solutions whereas for different data nonuniqueness appears. Therefore, a natural question arises: Where is the threshold between uniqueness and nonuniqueness of admissible weak solutions? We will review the history of results yielding the answer with one open problem still remaining. Further we take a look on the full Euler system and present some conjectures based on data observations.
  • 05.12.17   09:00

  • Tomasz Piasecki
  • (University of Warsaw)
  • On strong dynamics of a two component compressible mixture
  • Abstract:
  • The talk will be focused on the model of chemically reacting compressible mixture described by the Navier-Stokes system coupled with reaction-diffusion equations describing the evolution of mass fractions. The leading order term in these equations is so called flux diffusion matrix which can be in general non-symmetric. However, an appropriate change of unknowns leads to symmetrization of the system. Recently with Y. Shibata and E. Zatorska we have applied this procedure in case of a simple model with two species and constant temperature. In such case the symmetrization is particularly simple and can be shown directly which will be the first part of my talk. Then we rewrite the resulting symmetric system in Lagrangian coordinates and using estimates for the nonlinear part combined with the linear theory we show local existence of unique strong solutions. Finally we show an exponential decay estimate which, under additional smallness assumptions enables to show that our solution exists globally in time.
  • 28.11.17   09:00

  • Hind Al Baba
  • (Institute of Mathematics, CAS)
  • Maximal $L^p-L^q$ regularity to the Stokes problem with Navier or Navier-type boundary conditions
  • Abstract:
  • Stokes and Navier-Stokes equations play a central role in fluid dynamics, engineering and applied mathematics. Based on the theory of semigroups and on the complex and fractional power of operators we prove the maximal regularity of the Stokes Problem with the Navier or the Navier-type boundary conditions on the boundary of the fluid domain. These boundary conditions, while being perfectly motivated from the physical point of view, have been less studied than the most conventional Dirichlet boundary condition.
  • 21.11.17   09:00

  • Hana Mizerová
  • (Institute of Mathematics, CAS)
  • Global weak solutions to the kinetic Peterlin model
  • Abstract:
  • We consider a class of kinetic  models for viscoelastic fluids motivated by the Peterlin dumbbell theories for dilute polymer solutions with a nonlinear spring law for an infinitely extensible spring. The polymer molecules are suspended in an incompressible viscous Newtonian fluid confined to a bounded domain in two space dimensions. The unsteady motion of the solvent is described by the incompressible Navier-Stokes equations with the elastic extra stress tensor appearing as a forcing term in the momentum equation. The elastic stress tensor is defined by the Kramers expression through the probability density function of polymer molecules that satisfies the corresponding Fokker-Planck equation. We prove the existence of global-in-time weak solutions to the so-called kinetic Peterlin model.
    The present work has been done in collaboration with M. Lukáčová (Mainz), and A. ?wierczewska-Gwiazda, P. Gwiazda (Warsaw).
  • 14.11.17   09:00

  • Martin Michálek
  • (Institute of Mathematics, CAS)
  • Machine learning - recent breakthroughs and place for differential equations
  • Abstract:
  • Some recent advances in the field of machine learning will be summarized. Having them in mind, we will present the basic mathematical model for neural networks together with some of its properties. Some parallels and intersections with mathematical analysis and differential equations will be also discussed.
    (The lecture combines impressions from my participation in the Heidelberg Laureate Forum 2017. No prior knowledge of the field is assumed for the audience.)
  • 31.10.17   09:00

  • Václav Mácha
  • (Institute of Mathematics, CAS)
  • Global BMO estimates for non-Newtonian fluids with perfect slip boundary conditions
  • Abstract:
  • We study the generalized stationary Stokes system in a bounded domain in the plane equipped with perfect slip boundary conditions. We show natural stability results in oscillatory spaces, i.e. Hölder spaces and Campanato spaces including the border line spaces of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). Especially we show that under appropriate assumptions gradients of solutions are globally continues. Since the stress tensor is assumed to be governed by a general Orlicz function, our theory includes various cases of (possibly degenerate) shear thickening and shear thinning fluids; including the model case of power law fluids. It is a joint work with Sebastian Schwarzacher.
  • 24.10.17   09:00

  • Jiří Neustupa
  • (Institute of Mathematics, CAS)
  • A contribution to the theory of regularity of a weak solution to the Navier-Stokes equations via one component of velocity and other related quantities
  • Abstract:
  • We deal with a suitable weak solution (v,p) to the Navier-Stokes equations, where v=(v_1,v_2,v_3). We give a brief survey of known criteria of regularity that use assumptions on just one component of v. We show that the regularity of (v,p) at a space-time point (x_0,t_0) is essentially determined by the Serrin-type integrability of the positive part of a certain linear combination of v_1^2, v_2^2, v_3^2 and p in a backward neighborhood of (x_0,t_0). An appropriate choice of coefficients in the linear combination leads to the Serrin-type condition on one component of v or, alternatively, on the positive part of the Bernoulli pressure (1/2)|v|^2+p or the negative part of p, etc.
  • 17.10.17   09:00

  • Manuel Friedrich
  • (University of Vienna)
  • Carbon geometries as optimal configurations
  • Abstract:
  • Carbon nanostructures are identified with configurations of atoms interacting via empirical potentials. The specific geometry of covalent bonding in carbon is phenomenologically described by the combination of an attractive-repulsive two-body interaction and a three-body bond-orientation part. In this talk we investigate the strict local minimality of specific carbon configurations under general assumptions on the interaction potentials and discuss the stability of graphene, some fullerenes, and nanotubes. This is joint work with E. Mainini, P. Piovano, and U. Stefanelli.
  • 03.10.17   09:00

  • Aneta Wroblewska-Kamińska
  • (Imperial College, London)
  • The qualitative properties for viscous hydrodynamic models of collective behaviour with damping and nonlocal interactions
  • Abstract:
  • Hydrodynamic systems for interacting particles where attraction is taken into account by nonlocal forces derived from a potential and repulsion is introduced by local pressure arise in swarming modelling. We focus on the case where there is a balance between nonlocal attraction and local pressure in presence of confinement in the whole space. Under suitable assumptions on the potentials and the pressure functions, we show the global existence of solutions for these hydrodynamic models with viscosity and linear damping. By introducing linear damping into the system, we ensure the existence and uniqueness of compactly supported stationary densities with fixed mass and center of mass whose associated velocity field is zero in their support.  Moreover, we show that global weak solutions converge for large times to the set of these stationary solutions in a suitable sense.
    This is a joint result with José A. Carrillo.
  • 12.09.17   09:00

  • Joerg Wolf
  • (Chung-Ang University, Seoul)
  • On the energy concentration and self-similar blow of solutions to the Euler equations under the condition of Type I blow-up
  • Abstract:
  • See the attached file.
  • 30.08.17   10:20

  • Oliver Leingang
  • (Vienna University of Technology)
  • Discrete blow-up behaviour for the Keller-Segel system
  • Abstract:
  • In this talk I will give a short introduction to the Keller-Segel model and present new results addressing the existence of solution and the blow-up in finite time in the semi-discrete case. The Keller-Segel system is a macroscopic model which describes the collective motion of cells, usually bacteria or amoebas, guided by chemicals. The concept is widespread in nature and is called chemotaxis. One important outcome of this process is the aggregation of cells, which can lead to so called chemotactic collapse. Mathematically, this is described by a set of non-linear non-local equations for which the solutions show a dichotomy in the blow-up behaviour corresponding to the above mentioned chemotactic collapse. This blow-up dichotomy can be shown by a virial argument and my talk is concerned with the question of the existence of a numerical scheme which translates this continuous argument into the discrete case.
  •                    09:00

  • Antonín Novotný
  • (Université du Sud Toulon-Var)
  • On compressible Navier-Stokes system with non-zero inflow/outflow boundary conditions
  • Abstract:
  • We discuss the existence of weak solutions to the compressible evolutionary Navier-Stokes system with non-zero inflow/ouflow boundary conditions. The hard sphere pressure-density state equation is used to provide the necessary a priori bounds.
  • 08.08.17   09:00

  • Ansgar Jüngel
  • (TU Wien)
  • Nano processors: unimaginably small and unbelievably fast. Model hierarchy, analysis, simulations
  • Abstract:
  • The success of computer technology is mainly based on the miniaturization of the semiconductor devices in computer processors. As actual devices have a size of a few nanometers only, this process seems to reach its physical limit, and new technologies or materials are needed. In this talk, some aspects of the mathematical modeling of semiconductor devices, the analysis of the resulting nonlinear partial differential equations, and their numerical simulation will be presented. We focus on kinetic equations and their diffusion moment models, mixed finite-element and finite-volume approximations. In the analytical part, we detail a new technique, applied to a quantum diffusion equation. The new technique uses systematic integration by parts, which is based on a polynomial decision problem arising in real algebraic geometry and which allows for the derivation of a priori estimates.
  • 13.06.17   09:00

  • Piotr B. Mucha
  • (University of Warsaw)
  • A drop of water
  • Abstract:
  • I will talk about the issue of existence and uniqueness of solution to the Inhomogeneous Navier-Stokes system (INS) in a two dimensional domain. The key point it that the initial density can be just a characteristic  function of a set. Even for such rough solutions the result provides unique solutions. As a historical remark, it is emphasized that it solves the problem put by PL Lions in his book concerning weak solutions to INS.
    The talk will be based on joint results with Raphael Danchin (Paris). Preprint: https://arxiv.org/abs/1705.06061
  • 23.05.17   10:20

  • Jonas Sauer
  • (Max Planck Institute, Leipzig)
  • Time-Periodic L^p Estimates for Parabolic Boundary Value Problems
  • Abstract:
  • We introduce a method for showing a priori L^p estimates for time-periodic, linear, partial differential equations set in a variety of domains such as the whole space, the half space and bounded domains. The method is generic and can be applied to a wide range of problems. In the talk, I intend to demonstrate it on the Stokes equations and on parabolic boundary value problems. The latter example thus generalizes a famous result due to Agmon, Douglas and Nirenberg. The main idea is to replace the time axis with a torus in order to reformulate the problem on a locally compact abelian group and to employ Fourier analysis on this group. As a by-product, maximal L^p regularity for the corresponding initial-value problem follows for many operators such as the Dirichlet Laplacian and the Stokes operator without the notion of R-boundedness. In fact, we show that maximal L^p regularity for the initial value problem is even equivalent to time-periodic maximal L^p regularity.
    The talk is based on joint works with Yasunori Maekawa and Mads Kyed.
  •                    09:00

  • Šárka Nečasová
  • (Institute of Mathematics, CAS)
  • Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition
  • Abstract:
  • We shall consider the problem of the motion of a rigid body in an incompressible viscous fluid filling a bounded domain. This problem was studied by several authors. They mostly considered classical non-slip boundary conditions, which gave them very paradoxical result of no collisions of the body with the boundary of the domain. Only recently there are results when the Navier type of boundary are considered.

    We shall consider the Navier condition on the boundary of the body and the non-slip condition on the boundary of the domain. This case admits collisions of the body with the boundary of the domain. We shall prove the global existence of weak solution of the problem. Secondly, we prove local existence of strong solution and finally we will show weak-strong uniqueness.

    References:
    [1] Chemetov, Nikolai V., Necasova, Sarka: The motion of the rigid body in the viscous fluid including collisions. Global solvability result. Nonlinear Anal. Real World Appl. 34 (2017), 416-445
    [2] Chemetov, Nikolai V., Necasova, Sarka, Muha, Boris: Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition, Preprint 2017
  • 02.05.17   09:00

  • Petr Girg
  • (University of West Bohemia, Pilsen)
  • Asymptotically linear and Superlinear Systems of Elliptic PDEs - existence, multiplicity and bifurcation
  • Abstract:
  • We will summarize results for the Dirichlet problem for systems of Elliptic PDEs obtained jointly with Maya Chhetri in recent years. The first part of the talk will be focused on asymptotically linear problems and bifurcation from infinity at an eigenvalue. Our aim is to establish bifurcation of positive and negative solution from infinity. We will discuss Lyapunov-Schmidt method and obtain Landesman-Lazer type condition for systems.  In the case of a particular type of 3X3 system we obtain surprising result of bifurcation from infinity of positive solutions at no/one/ or two eigenvalues. The second part of the talk will be focused on superlinear problems. We will briefly introduce rescalling method which is suitable for studying equations with nonlinearities with asymptotically power type growth. We will discuss several concepts of subritical growth known from the literature. Finaly, we will present recent results for superlinear nonlinearities involving supercritical growth. These results were obtained using approximation of the superlinear problem by a sequence of asymptotically linear problems.

    Based on the following series of papers:

    Asymptotically linear problems:
    MR3548275 Chhetri, Maya ; Girg, Petr . Asymptotically linear system of three equations near resonance. J. Differential Equations 261 (2016), no. 10, 5900--5922.
    MR3504015 Chhetri, Maya ; Girg, Petr . On the solvability of asymptotically linear systems at resonance. J. Math. Anal. Appl. 442 (2016), no. 2, 583--599.
    MR3280138 Chhetri, Maya ; Girg, Petr . Asymptotically linear systems near and at resonance. Bound. Value Probl. 2014, 2014:242, 21 pp. 
    Superlinear problems:
    MR3548268 Chhetri, M. ; Girg, P. Global bifurcation of positive solutions for a class of superlinear elliptic systems. J. Differential Equations 261 (2016), no. 10, 5719--5733.
    MR3085072 Chhetri, Maya ; Girg, Petr . Existence of positive solutions for a class of superlinear semipositone systems. J. Math. Anal. Appl. 408 (2013), no. 2, 781--788.
    MR2548730 Chhetri, Maya ; Girg, Petr . Existence and nonexistence of positive solutions for a class of superlinear semipositone systems. Nonlinear Anal. 71 (2009), no. 10, 4984--4996.
  • 25.04.17   09:00

  • Sebastian Schwarzacher
  • (Charles University)
  • On compressible fluids interacting with a linear-elastic Koiter shell
  • Abstract:
  • We study the Navier--Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter's elastic energy.

    We discuss the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies $gamma>frac{12}{7}$ ($gamma>1$ in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in [D. Lengeler, M. Ruzicka, Weak Solutions for an Incompressible Newtonian Fluid Interacting with a Koiter Type Shell. Arch. Ration. Mech. Anal. 211 (2014), no. 1, 205--255] on incompressible Navier--Stokes equations.

    It is a joint work with D. Breit (Heriot-Watt Univ. Edinburgh).
  • 18.04.17   09:00

  • Tomasz Piasecki
  • (University of Warsaw)
  • On the stationary flow of a reactive gaseous mixture
  • Abstract:
  • We are interested in a system of equations describing stationary flow of a mixture of gases undergoing reversible chemical reactions. The system consists of the stationary compressible Navier-Stokes-Fourier equations coupled with reaction-diffusion equations describing balance of fractional masses. In particular we admit strong cross-diffusion, however we assume that molar masses of all species are equal.

    We show existence of weak solutions using the new pressure estimates developed recently for the compressible Navier-Stokes system. It enables to extend the range of gamma in the pressure law strengthening the previous existence results for the system under consideration. We also introduce a slightly more general notion of variational entropy solutions (which are weak solutions as long as the latter exist) and show existence of this type of solutions for gamma>1. This is a joint work with Milan Pokorny.
  • 11.04.17   09:00

  • Patrick Dondl
  • (University of Freiburg)
  • The Effect of Forest Dislocations on the Evolution of a Phase-Field Model for Plastic Slip
  • Abstract:
  • We consider a phase field model for dislocations introduced by Koslowski, Cuitino, and Ortiz in 2002. The model describes a single slip plane and consists of a Peierls potential penalizing non-integer slip and a long range interaction modeling elasticity. Forest dislocations are introduced as a restriction to the allowable phase field functions: they have to vanish at the union of a number of small disks in the plane. Garroni and Müller proved large scale limits of these models in terms of Gamma-convergence, obtaining a line-tension energy for the dislocations and a bulk term penalizing slip. This bulk term is a capacity stemming from the forest dislocations.

    In the present work, we show that the contribution of the forest dislocations to the the viscous gradient flow evolution is small. In particular it is much slower than the timescale for other effects like elastic attraction/repulsion of dislocations, which, by a recent result due to del Mar Gonzales and Monneau is already slower than the time scale from line tension energy. Overall, this leads to an effective behavior like a gradient flow in a wiggly potential.
  • 28.03.17   09:00

  • Hind Al Baba
  • (Institute of Mathematics, Czech Academy of Sciences)
  • The Oseen resolvent problem with Navier and full slip boundary conditions in Lp spaces
  • Abstract:
  • We consider the resolvent of the Oseen problem with the Navier boundary conditions in $L^p$ spaces and we prove a resolvent estimate for the solution to the corresponding problem.
  • 21.03.17   09:00

  • Jiri Neustupa
  • (Institute of Mathematics, Czech Academy of Sciences)
  • On the structure of the solution set of steady equations of motion of a class of non-Newtonian fluids
  • 07.03.17   09:00

  • Vaclav Macha
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Holder continuity of velocity gradients for shear-thinning fluids
  • Abstract:

  • We deal with a non-stationary flow of shear-thinning fluid in a bounded domain with perfect slip boundary condition. We provide a proof of the existence of a solution which is Holder continuous. This is a joint work with Jakub Tichy.

  • 28.02.17   09:00

  • Dominic Breit
  • (Heriot-Watt University, Edinburgh)
  • Singular limits for compressible fluids with stochastic forcing
  • Abstract:

  • I will present new results on stochastic Navier-Stokes equations for compressible fluids. I will introduce the concept of "finite energy weak martingale solutions". These solutions are weak in the analytical sense and in the probabilistic sense as well. In addition, they allow to control the evolution of the total energy. So, we can study the asymptotic behavior of the problem. In particular, I will identify the stochastic incompressible Navier-Stokes equations (Euler equations) as target system if the Mach number (and the viscosity) tends to zero.

  • 24.01.17   09:00

  • Antonín Novotný
  • (Université de Toulon)
  • Low Mach asymptotics for some numerical schemes for compressible Navier-Stokes equations
  • Abstract:
  • We investigate error between any discrete solution of several numerical schemes for compressible Navier-Stokes equations in low Mach number regime and an exact strong solution of the incompressible Navier-Stokes equations. The main tool is a discrete version of the relative energy method. We get unconditional error estimate in terms of explicitly determined positive powers of the space-time disretization parameters and Mach number in the case of well prepared initial data, and the boundedness of the error if the initial data are ill prepared. The  multiplicative constant in the error estimates depends on the suitable norm of the strong solution but is independent on the numerical solution itself (and of course, on the disretization parameters and the Mach number).
  • 10.01.17   10:20

  • Pierre-Étienne Druet
  • (WIAS Berlin)
  • Existence of weak solutions for improved Nernst-Planck-Poisson models of compressible electrolytes
  • Abstract:
  • The main components of a battery are two electrodes that are immersed in some electrolyte. The essential processes determining the performance of the battery happen in the neighborhood of the electrode-electrolyte contact. Here the classical models of Nernst-Planck-Poisson type completely fail to adequately predict the relevant phenomena. The thermodynamically correct coupling of diffusion, adsorption and reactions at interfaces particularly requires a coupled system of the Poisson equation, reaction diffusion equations and the compressible Navier-Stokes equations. In this talk we will present a recent existence result for this system, including the crucial pressure contribution in the diffusion fluxes.

     

  •                    09:00

  • Tomáš Dohnal
  • (TU Dortmund)
  • Rigorous Asymptotics of Moving Pulses for Nonlinear Wave-Problems in Periodic Structures
  • Abstract:

  • The possibility of moving, spatially localized pulses of constant or time periodic form in periodic media, e.g. in photonic crystals, is interesting from the mathematical as well as the applied side. An example is optical computing where such pulses could function as bit carriers.

    Pulses in the form of asymptotically small and wide wavepackets can be studied with the help of envelope approximations. Hereby the envelope satisfies an effective equation with constant coefficients. We discuss rigorous results of such approximations in one spatial dimension on long time intervals for a nonlinear wave equation and a nonlinear Schrödinger equation. We concentrate on the asymptotic scaling which leads to the, so called, coupled mode equations (CMEs) of first order. CMEs have families of solitary waves parametrized by velocity, such that in the original model propagation of localized pulses is possible for a range of velocities at one fixed frequency. The justification proof relies on the Bloch transformation, Sobolev space estimates and the Gronwall inequality. Besides the idea of the proof we present also some numerical examples.

  • 03.01.17   09:00

  • Didier Bresch
  • (LAMA, Université Savoie Mont Blanc, CNRS )
  • Some news concerning the compressible Euler-Korteweg system
  • 13.12.16   09:00

  • Jürgen Sprekels
  • (Humboldt University Berlin and WIAS Berlin )
  • On a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition
  • 06.12.16   09:00

  • Nicola Zamponi
  • (TU Wien)
  • Existence analysis of a single-phase flow mixture with van der Waals pressure
  • Abstract:

  • The transport of single-phase fluid mixtures in porous media is described by cross-diffusion equations for the chemical concentrations. The equations are obtained in a thermodynamic consistent way from mass balance, Darcy's law, and the van der Waals equation of state for mixtures. Including diffusive fluxes, the global-in-time existence of weak solutions in a bounded domain with equilibrium boundary conditions is proved, using the boundedness-by-entropy method. Based on the free energy inequality, the large-time convergence of the solution to the constant equilibrium concentration is shown. For the two-species model and specific diffusion matrices, an integral inequality is proved, which reveals a maximum and minimum principle for the ratio of the concentrations. Without diffusive fluxes, the two-dimensional pressure is shown to converge exponentially fast to a constant. Numerical examples in one space dimension illustrate this convergence.

  • 29.11.16   09:00

  • Milan Pokorný
  • (Charles University, Prague)
  • Incompressible fluid model of electrically charged chemically reacting and heat conducting mixtures
  • Abstract:

  • We study a model of a mixture of fluids which is modeled by an incompressible nonNewtonian
    (power-law) fluid. We allow that the constituents may undergo chemical
    reactions and the fluid in total can transfer heat and is generally electrically charged.
    We show existence of a weak solution to the corresponding system of partial differential
    equations which exists globally in time and without any restriction on the size of the
    data. It is a joint work with Miroslav Bulíček (Charles University in Prague) and Nicola
    Zamponi (Vienna University of Technology).

  • 22.11.16   09:00

  • Šárka Nečasová
  • (Institute of Mathematics, CAS)
  • Rigorous derivation of the equations describing objects called "accretion disk"
  • Abstract:
  • See attachment
  • 15.11.16   09:00

  • Reinhard Farwig
  • (TU Darmstadt)
  • Almost optimal initial value conditions for the Navier-Stokes equations: existence, uniqueness, continuity, and stability
  • Abstract:
  • See the attachment.
  • 08.11.16   09:00

  • Matteo Caggio
  • (Institute of Mathematics, CAS)
  • Regularity criteria for the Navier-Stokes equations based on one component of velocity
  • Abstract:
  • We study the regularity criteria for the incompressible Navier-Stokes equations in the whole space R^3 based on one velocity component, namely u_3, nabla u_3 and nabla^2 u_3. We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition nabla u_3 in L^beta (0,T;L^p), where 2/beta + 3/p = 7/4 + 1/(2p) and p in (2,infty], yields the regularity of u on (0,T]. 
  • 18.10.16   09:00

  • Boris Muha
  • (University of Zagreb)
  • An operator splitting scheme for the fluid structure interaction problems
  • Abstract:

  • Motivation for studying fluid-structure interactions (FSI) problems comes from applications in various areas including geophysics, biomedicine and aeroelasticity. The FSI problems are typically nonlinear systems of the partial differential equations of parabolic-hyperbolic type with the moving boundary. In this talk we will present an operator splitting numerical scheme, so-called the kinematically coupled scheme, for the FSI problems and show how ideas from the numerical scheme can be be used in the constructive proof of the existence of a weak solution for various FSI problems. We will also discuss the rate of convergence of the numerical scheme and its extensions. 
    The presented results are part of joined work with S. Canic, University of Houston and M. Bukac, University of Notre Dame.

  • 11.10.16   09:00

  • Anja Schlomerkemper
  • (University of Wurzburg )
  • Existence of weak solutions to an evolutionary model of magnetoelastic materials
  • Abstract:

  • The evolutionary model for magnetoelasticity that we consider is phrased in Eulerian coordinates. It is a system of partial differential equations that contains (1) a Navier-Stokes equation with magnetic and elastic terms in the stress tensor obtained by a variational approach, (2) a regularized transport equation for the deformation gradient and (3) the Landau-Lifshitz-Gilbert equation for the dynamics of the magnetization. The proof of existence of weak solutions is based on a Galerkin method and a fixed-point argument combined with ideas from the analysis of models for the flow of liquid crystals (F.-H. Lin and C. Liu) and of the Landau-Lifshitz equation (G. Carbou and P. Fabrie).

  • 04.10.16   09:00

  • Karolina Weber
  • (TU Wien)
  • A Stochastic Reaction-Diffusion Model with Multiplicative Noise
  • Abstract:

  • We consider stochastic reaction-diffusion equations with a multiplicative noise term and analyse the influence of the Brownian Motion on the solution. Therefore, we use a variational approach to show the existence of solutions for a competition model for two species. Moreover, numerical simulations will be presented for the stochastic model and compared to the deterministic case.

  • 21.09.16   09:45

  • Anita Gerstenmayer
  • (TU Wien)
  • A cross-diffusion model for ion transport
  • Abstract:

  • Ion transport can be modelled using the Poisson-Nernst-Planck (PNP) equations. In order to account for size exclusion effects in narrow ion channels, the PNP model can be modified leading to a cross-diffusion system. In this talk, the modified PNP model and some analytic and numerical results will be presented. It will be discussed how an entropy method can be applied to prove the global-in-time existence of weak solutions to the model. Furthermore, a finite volume discretization of the equations and some simulation results for a calcium- selective channel will be shown.

  •                    09:00

  • Konstantin Pileckas
  • (Vilnius University)
  • Stationary Navier-Stokes equations with nonhomogeneous boundary conditions in 2D symmetric unbounded domains
  • Abstract:

  • The stationary nonhomogeneous Navier-Stokes problem is studied in a two dimensional symmetric domain with a semi-infinite outlet to infinity (for instance, paraboloid type or channel-like). Under the symmetry assumptions on the domain, boundary values and external force the existence of at least one weak symmetric solution is proved without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value over the inner and the outer boundaries may be arbitrarily large. The Dirichlet integral of the solution can be finite or infinite dependent on the geometry of the domain.

  • 01.06.16   09:00

  • Julian Fischer
  • (Max Planck Institute, Leipzig)
  • A higher-order large-scale regularity theory for random elliptic operators
  • Abstract:

  •  

    We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields $a$ in the context of stochastic homogenization. Under the assumptions of stationarity and slightly quantified ergodicity of the ensemble, we derive a $C^{2,alpha}$-``excess decay'' estimate on large scales and a $C^{2,alpha}$-Liouville principle: For a given $a$-harmonic function $u$ on a ball $B_R$, we show that its energy distance to the space of $a$-harmonic ``corrected quadratic polynomials'' on some ball $B_r$ has the natural decay in the radius $r$ above some minimal (random) radius $r_0$. Our Liouville principle states that the space of $a$-harmonic functions growing at most quadratically has (almost surely) the same dimension as in the constant-coefficient case. The existence of $a$-harmonic ``corrected quadratic polynomials'' -- and therefore our regularity theory -- relies on the existence of second-order correctors for the homogenization problem. By an iterative construction, we are able to establish existence of subquadratically growing second-order correctors. This is a joint work with Felix Otto.

  • 24.05.16   09:00

  • Jesus Hernandez
  • (Universidad Complutense, Madrid )
  • Linearized stability for solutions to nonlinear degenerate and singular parabolic problems
  • Abstract:

  • We study linear eigenvalue problems with singular (unbounded close to the boundary) coefficients arising in the linearization to positive solutions to some degenerate and singular problems. This improves previous work by Bertsch and Rostamian and Hernandez, Mancebo and Vega. This is related with solutions to the linear Schrödinger equation and compact solutions for some associated nonlinear problems.

    Joint work with J.I.Diaz.

  • 17.05.16   10:20

  • Goro Akagi
  • (Tohoku University; Helmholtz Zentrum München; TU München)
  • Stability analysis of non-isolated asymptotic profiles for fast diffusion
  • Abstract:

  •  

    This talk is concerned with asymptotic profiles for solutions to the Cauchy-Dirichlet problem
    for the Fast Diffusion equation (FD) in smooth bounded domains under the so-called Sobolev
    subcritical condition. It is well-known that every solution of (FD) vanishes in finite time
    with a power rate; more precisely, it asymptotically approaches to a separable solution
    (Berryman and Holland '80). Then the asymptotic profile for each vanishing solution can be
    characterized as a non-trivial solution of the Emden-Fowler equation (EF). The stability of
    asymptotic profiles has been discussed for the case that (EF) has a unique positive solution;
    on the other hand, the case that (EF) may have multiple (positive) solutions had not been
    studied for many years.

    In this talk, we shall first see how to formulate notions of stability and instability of asymptotic
    profiles, and then, we shall discuss criteria to distinguish the stability of each asymptotic profile.
    Moreover, we shall focus on how to treat non-isolated asymptotic profiles; indeed, (EF) may admit a
    one-parameter family of positive solutions, e.g., for sufficiently thin annular domains. In particular,
    for thin annular domain cases, each non-radial asymptotic profile belonging to a one-parameter family
    turns out to be stable and the radial positive profile turns out to be unstable. The method of analysis
    relies on variational method, uniform extinction estimates for solutions to (FD), the Lojasiewicz-Simon
    inequality and energy techniques developed for doubly nonlinear evolution equations.

  •                    09:00

  • Ansgar Jüngel
  • (TU Wien)
  • Multi-species systems in biology: cross-diffusion and hidden gradient-flow structure
  • Abstract:

  •  

    Nature is dominated by systems composed of many individuals with a collective
    behavior. Examples include wildlife populations, biological cell dynamics, and
    tumor growth. There is a fast growing interest in multi-species systems both in
    theoretical biology and applied mathematics, but because of their enormous complexity,
    the scientific understanding is still very poor. On a macroscopic level, such systems
    may be modeled by systems of partial differential equations with cross diffusion,
    which reveals surprising effects such as uphill diffusion and diffusion-induced
    instabilities, seemingly contradicting our intuition on diffusion.

    Major difficulties of the mathematical analysis of the cross-diffusion equations
    are their highly nonlinear structure and the lack of positive definiteness of the
    diffusion matrix. In this talk, a method inspired from non-equilibrium thermodynamics
    is proposed, which allows for a mathematical theory of a large class of such systems.
    The idea is to exploit the hidden formal gradient-flow structure by introducing
    so-called entropy variables. The analysis in these variables leads to global
    existence results, L^infty bounds, and large-time asymptotics results.
    We apply the technique to some systems modeling populations and tumor growth.

  • 10.05.16   09:00

  • Hind Al Baba
  • (Institute of Mathematics, CAS)
  • Semi-groups theory for the Stokes and Navier-Stokes equations with Navier-type boundary conditions
  • Abstract:

  • Since the pioneer work of Leray and Hopf, Stokes and Navier-Stokes problems
    have been often studied with Dirichlet boundary condition. Nevertheless, in the
    opinion of engineers and physicists such a condition is not always realistic in industrial
    and applied problems of origin. Thus arises naturally the need to carry out a
    mathematical analysis of these systems with different boundary conditions, which
    best represent the underlying fluid dynamic phenomenology. Based on the theory
    of semi-groups we carry out a systematic treatment of Stokes and Navier-Stokes
    equations with Navier or Navier-type boundary conditions and boundary conditions
    involving the pressure in L^p-spaces. These boundary conditions are usually
    called in the literature, non-standard boundary conditions on the boundary of the
    fluid domain.

  • 03.05.16   09:00

  • Šimon Axmann
  • (Charles University in Prague)
  • Strong solutions to the steady Navier-Stokes system for dense compressible ?uids
  • Abstract:

  • We study the existence of strong solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity under the additional assumption that the fluid is sufficiently dense. The investigation is connected to the corresponding singular limit as Mach number goes to zero.

     

  • 26.04.16   09:00

  • Pavel Drábek
  • (University of West Bohemia, Pilsen)
  • Convergence to travelling waves in the Fisher-Kolmogorov equation with a non-Lipschitzian reaction term
  • Abstract:

  • We consider the semilinear Fisher-Kolmogorov-Petrovski-Piscounov equation for the advance
    of an advantageous gene in biology. Its nonsmooth reaction function f(u) allows for the introduction of
    travelling waves with a new pro file. We study existence, uniqueness, and long-time asymptotic behavior
    of the solutions of the initial value problem to a travelling wave.

  • 19.04.16   09:00

  • Petr Stehlík
  • (University of West Bohemia, Pilsen)
  • Qualitative properties of lattice reaction-diffusion equations
  • Abstract:

  • In this talk we discuss basic properties of diffusion and reaction-diffusion equations on lattices. Formulating the problem as an abstract diff. equation in sequence spaces we show existence, uniqueness and continuous dependence on the initial condition as well as the convergence of the discretized reaction-diffusion equation. We conclude with maximum principles and a priori estimates. (joint work with Antonín Slavík and Jonáš Volek)

  • 12.04.16   09:00

  • Bernard Ducomet
  • (CEA DAM DIF)
  • Strong solution with critical regularity of a polytropic model of radiating flow
  • Abstract:

  • We aim at investigating the physically relevant situation of polytropic flows. More precisely, we consider a model arising in radiation hydrodynamics which is based on the full Navier-Stokes-Fourier system describing the macroscopic fluid motion, and a P 1-approximation of the transport equation modeling the propagation of radiative intensity. In the strongly under-relativistic situation, we establish the global-in-time existence and uniqueness of solutions with critical regularity for the associated Cauchy problem with initial data close to a stable radiative equilibrium. We also justify the non-relativistic limit in that context. For smoother (possibly) large data bounded away from the vacuum and more general physical coefficients that may depend on both the density and the temperature, the local existence of strong solutions is shown.

  • 05.04.16   09:00

  • Martin Michálek
  • (Institute of Mathematics, CAS)
  • Existence of global weak solutions for compressible Navier-Stokes system with Entropy Transport
  • Abstract:

  • Compressible Navier-Stokes system with Entropy Transport serves as a simplified model for the compressible heat conducting fluid. A former result on stability of solutions to the mentioned system is extended on an existence result by giving a suitable approximative scheme. There are two formally equivalent formulations of the equation for the entropy, namely pure transport equation for the entropy $s$ and continuity equation for the entropy density $varrho s$ (where $varrho$ is the density). A crucial role in the existence part plays the possibility to switch between these formulations even in the case of weak solutions.

  • 29.03.16   09:00

  • Elisa Davoli
  • (University of Vienna)
  • Homogenization of integral energies under periodically oscillating differential constraints
  • Abstract:

  • We present a homogenization result for a family of integral energies, where the field under consideration are subjected to periodically oscillating differential constraints in divergence form. The work is based on the theory of A-quasiconvexity with variable coefficients and on two-scale techniques.

  • 22.03.16   09:00

  • Petr Girg
  • (University of West Bohemia, Plzeň)
  • Nonuniqueness of solutions of initial-value problems for parabolic p-Laplacian and multi-bump solutions
  • Abstract:
  • We will consider a quasilinear parabolic problem with the p-Laplacian and a non-Lipschitz reaction function 
    and we will discuss     nonuniqueness for zero initial data. Our method is based on sub- and supersolutions and 
    the weak comparison principle. Moreover, for p>2, we use     Barenblatt type functions as supersolutions 
    to obtain     nonnegative multi-bump solutions with spatially disconnected compact supports. The presented 
    results are joint work with     Jiri Benedikt, Vladimir E. Bobkov, Lukas Kotrla, and Peter Takac.
  • 15.03.16   09:00

  • Radim Hošek
  • (Institute of Mathematics, CAS)
  • Finite difference scheme to compressible Navier-Stokes equations
  • Abstract:
  • Inspired by works of Karper, Feireisl and their co-authors, we propose a finite difference scheme to the system of compressible Navier-Stokes equation in three spatial dimension and show the (first half of the way to proving) convergence of the numerical solutions to a weak solution of the system. The difficulties that occur both when generalizing from 1D as well as acommodating the theory developed for FEM-DG using Crouzeix-Raviart elements will be pointed out.
  • 01.03.16   09:00

  • Ondřej Kreml
  • (Institute of Mathematics, CAS)
  • On measure valued solutions to the compressible Euler equations
  • Abstract:

  • In a very interesting paper, Szekelyhidi and Wiedemann (2012) proved that every measure valued solution to the incompressible Euler equations can be approximated by a sequence of weak solutions, implying that there is no significant difference between weak and measure valued solutions to the incompressible Euler system. In this talk we prove that such a property does not hold for the compressible case and we show the construction of a measure valued solution which can not be generated by weak solutions. Moreover we show an abstract neccesary condition for measure valued solutions to be generated by sequences of weak solutions. The proof is based on a generalization of a rigidity result by Ball and James, the necessary condition is obtained as a consequence of the works of Fonseca and Muller. We present also some connections between the compressible Euler system and problems of gradient Young measures arising in nonlinear elasticity.
    This is a joint work with Elisabetta Chiodaroli, Eduard Feireisl and Emil Wiedemann.

  • 12.01.16   09:00

  • Eduard Feireisl
  • (Institute of Mathematics, CAS)
  • Measure-valued solutions to compressible Navier-Stokes/Euler systems revisited
  • Abstract:

  • We introduce a new concept of a dissipative measure valued solution to the compressible Navier-Stokes/Euler system based on the quantity called dissipation defect. We identify a large class of problems including certain numerical schemes generating dissipative measure valued solutions. Finally, we show uniqueness of strong solutions in the class of measure valued solutions (weak-strong uniqueness) and characterize bounded-density measure valued solutions to the Navier-Stokes system. Applications to convergence problems will be given. This is a joint work with P.Gwiazda, A.Swierczewska-Gwiazda, and Emil Wiedemann.

     

  • 05.01.16   09:00

  • Camillo De Lellis
  • (Universität Zürich )
  • Approaching Plateau's problem with minimizing sequences of sets
  • Abstract:
  • In a joint paper with Francesco Ghiraldin and Francesco Maggi we provide a compactness
    principle which is applicable to different formulations of Plateau's problem in codimension one
    and which is exclusively based on the theory of Radon measures and elementary comparison
    arguments. Exploiting some additional techniques in geometric measure theory, we can use
    this principle to give a different proof of a theorem by Harrison and Pugh and to answer a
    question raised by Guy David.
  • 15.12.15   09:00

  • Milan Pokorny
  • (Charles University in Prague )
  • Heat-conducting, compressible mixtures with multicomponent diffusion
  • 08.12.15   09:00

  • Tomasz Piasecki
  • (Institute of Mathematics of the Polish Academy of Sciences)
  • Stationary compressible Navier-Stokes Equations with inflow boundary
  • Abstract:
  • The existence of solutions to the stationary compressible Navier-Stokes equations with boundary conditions admitting inflow and outflow is in general open question. All known results require some assumptions on smallness of the data and additional conditions on the shape of the boundary. I will discuss briefly known results of this type and show a new estimate in fractional order Sobolev spaces which seems new and promising approach in stationary case. This is a joint result with Piotr Mucha.
  • 01.12.15   09:00

  • Jiri Jarusek
  • (IM CAS)
  • Rational contact model with finite interpenetration
  • Abstract:
  • The classical Signorini contact model respects the impenetrability of Mass. However, the microscopic structure of every material even that with macroscopically seemingly perfect surface has small deformable asperities and/or holes to be filled. The model to be presented allows some surface interpenetration between the body and the contacted support. Unlike "normal compliance" approach sometimes used, the interpenetration is strictly limited here and its limit is not reachable outside a zero measure. The static version of the model for both frictionless and frictional contact will be treated together with the dynamic frictionless version.
  • 24.11.15   09:00

  • Jiří Neustupa
  • (IM CAS)
  • A spectral criterion for stability of a steady flow in an exterior domain
  • Abstract:
  • We show that the question of stability of a steady incompressible Navier-Stokes flow V in a 3D exterior domain depends on the time-decay of a finite family of concretely defined functions. Then, although the associated linearized operator has an essential spectrum touching the imaginary axis, we show that certain assumptions on the eigenvalues of this operator guarantee the stability of flow V, regardless the essential spectrum. No assumption on the smallness of the steady flow V is required.
  • 10.11.15   09:00

  • Filippo Dell'Oro
  • (IM CAS)
  • Energy inequality in differential form for weak solutions to the compressible Navier-Stokes equations on unbounded domains
  • Abstract:
  • We consider the Navier-Stokes equations of compressible isentropic viscous fluids on an unbounded three-dimensional domain with a compact Lipschitz boundary. Under the condition that the total mass of the fluid is finite, we show the existence of globally defined weak solutions satisfying the energy inequality in differential form. This is a joint work with E. Feireisl.
  • 03.11.15   09:00

  • Margarida Baía
  • (Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal)
  • A model for phase transitions with competing terms
  • 20.10.15   10:20

  • Elisabetta Chiodaroli
  • ( École Polytechnique Fédérale de Lausanne  )
  • A class of large global solutions for the Wave-Map equation
  • Abstract:
  • In this talk we consider the equation for equivariant wave maps from
    3+1- Minkowski space-time to the three dimensional sphere and we prove
    global in forward time existence of certain smooth solutions which have
    infinite critical Sobolev norm. Our method is based on a perturbative
    approach around suitably constructed approximate self-similar
    solutions.
  •                    09:00

  • Emil Wiedemann
  • (Hausdorff Center for Mathematics)
  • Weak-strong uniqueness in fluid dynamics
  • 13.10.15   10:20

  • Agnieszka Swierczewska-Gwiazda
  • (Faculty of Mathematics, Informatics and Mechanics, University of Warsaw)
  • On various compressible models of fluid mechanics: weak and measure-valued solutions
  • Abstract:
  • I will discuss the issue of existence of weak and measure-valued solutions to various systems of Euler type.  The most attention will be directed to the system of shallow water  type capturing flows of granular media, but I will also mention the pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity appearing in  collective behavior patterns.
  •                    09:00

  • Water Supply Interruption
  • Seminar is cancelled
  • 06.10.15   09:00

  • Filip Rindler
  • (University of Warwick)
  • Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms
  • Abstract:
  • !!! EXCEPTIONALLY AT SEMINAR ROOM K6, FACULTY OF MATHEMATICS AND PHYSICS, CHARLES UNIVERSITY IN PRAGUE, SOKOLOVSKA 83, PRAHA 8 !!!

    Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^p-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending both the theory of (generalized) Young measures and the theory of H-measures. Since in L^p-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, MCFs allow to quantify the difference between these two notions of compactness. The definition involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily.
    Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. Moreover, the hierarchy of oscillations with regard to slow and fast scales can be investigated as well since this information is also is reflected in the generated MCF.
  • 02.06.15   09:00

  • Vaclav Macha
  • Dynamics of a body containing a vicsous compressible fluid
  • 19.05.15   11:15

  • Jiri Neustupa
  • Stability of a steady flow of a viscous incompressible fluid past a fixed or rotating body
  •                    10:40

  • Patrick Penel
  • Mathematical gardening: two tools recently used in the theory of Navier-Stokes equations
  •                    09:40

  • Miloslav Feistauer
  • Analysis of discontinuous Galerkin method for PDEs with corner singularities
  •                    09:05

  • Petr Kaplicky
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On $L^q$ estimates of planar flows up to boundary
  • Abstract:
  • We show apriori $L^q$ gradient estimates for a sufficiently smooth planar flow driven by generalized Stokes system of equations. The estimates are obtained up to the boundary of a container where the fluid is contained. We allow power growth $p-1$, $pin(1,+infty)$ of the extra stress tensor for large values of shear rate but we exclude degeneracy or singularity for small shear rate. We also allow arbitrary $qin[p,+infty)$. The technique is based on a new type of Sobolev embedding theorem.
  • 12.05.15   09:00

  • Martin Väth
  • (Free University Berlin)
  • Stability in L_2, W^{1,2}: Extrapolation Spaces and Gel'fand Triples
  • Abstract:
  • Starting from the "naive" question how to prove stability of semilinear parabolic systems in the spaces L_2 and W^{1,2} by means of linearization, one is led to two different "natural" approaches. The aim of the talk is to compare both approaches and to reveal some connections, e.g. with Kato's square root problem.
  • 05.05.15   09:00

  • Simon Axmann
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On a result of D. Gerard-Varet and N. Masmoudi in homogenization
  • Abstract:
  • We will refer on the paper of David Gerard-Varet and Nader Masmoudi: Homogenization and boundary layers
  • 21.04.15   10:20

  • Jan Burczak
  • (Institute of Mathematics, Polish Academy of Sciences)
  • The Keller-Segel system
  • Abstract:
  • I will present one of the most popular PDEs stemming from the mathematical biology, namely the Keller-Segel system. The first part of my talk is devoted to presenting applicational motivation and classical rudiments of the analysis of the Keller-Segel system. In the second part, I will focus on Keller-Segel systems with general diffusions, including semilinear and fractional ones. In particular, a disproof of a blowup conjecture for the critical, fractional one-dimensional case will be sketched. This last part is based on a joint work with Rafael Granero (Davies).
  •                    09:00

  • Simon Axmann
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On a result of D. Gerard-Varet and N. Masmoudi in homogenization
  • Abstract:
  • We will start referring on the paper of David Gerard-Varet and Nader Masmoudi: Homogenization and boundary layers
  • 14.04.15   10:20

  • Radim Hosek
  • (Institute of Mathematics, Academy of Sciences of the Czech Republic)
  • Strongly regular family of meshes approximating C^2 domains for numerical method for compressible Navier-Stokes
  • Abstract:
  • In order to establish error estimates for a numerical method to compressible Navier-Stokes equations, we need a family of tetrahedral meshes covering approximative polyhedral domains $Omega_h$ satisfying $dist[partial Omega_h, partial Omega] < c h^2$, where $Omega$ is at least $C^2$ smooth bounded domain in 3 dimensional space.
    We show that having an initial mesh with certain properties, we can construct a strongly regular family of meshes.
  •                    09:00

  • Simon Axmann
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On a result of D. Gerard-Varet and N. Masmoudi in homogenization
  • Abstract:
  • We will start referring on the paper of David Gerard-Varet and Nader Masmoudi: Homogenization and boundary layers.
  • 07.04.15   10:50

  • Manoj K. Yadav
  • (Faculty of Civil Engeneering, Czech Technical University in Prague)
  • On a result of N. Masmoudi in homogenization
  • Abstract:
  • We will finish referring on the paper of Nader Masmoudi: Homogenization of the compressible Navier-Stokes equations in a porous medium.
  •                    10:20

  • Sarka Necasova
  • (Institute of Mathematics, Academy of Sciences of the Czech Republic)
  • Singular limits in a model of compressible flow
  • Abstract:
  • We consider relativistic and "semi-relativistic" models of radiative viscous compressible Navier-Stokes-(Fourier) system coupled to the radiative transfer equation extending the classical model introduced in [1] and we study some of its singular limits in the case of well-prepared initial data and Dirichlet boundary condition for the velocity field see [2], [3], [4].

    References
    [1] B. Ducomet, E. Feireisl, S. Necasova: On a model of radiation hydrodynamics. Ann. I. H. Poincare - AN 28 (2011) 797–812.
    [2] B. Ducomet, S. Necasova: Diffusion limits in a model of radiative flow, to appear in Annali dell Universita di Ferrara, DOI 10.1007/s11565-014-0214-3.
    [3] B. Ducomet, S. Necasova: Singular limits in a model of radiative flow, to appear in J. of Math. Fluid Mech.
    [4] B. Ducomet, S. Necasova: Non equilibrium diffusion limit in a barotropic radiative flow, submitted to Volume Contemporary Mathematics Series of the American Mathematical Society; editors: Vicentiu Radulescu, Adelia Sequeira, Vsevolod A. Solonnikov.
  •                    09:00

  • Tomasz Piasecki
  • (Institute of Mathematics, Polish Academy of Sciences)
  • Stationary compressible Navier-Stokes equations with inflow boundary conditions
  • Abstract:
  • I will discuss recent results in the theory of compressible flows described by Navier-Stokes equations focusing on problems with inhomogeneous boundary data admitting inflow and outflow through the boundary. Admission of inflow /outflow leads to substantial mathematical difficulties. In particular there are no general existence results in the stationary case in such situation what motivates investigation of small data problems and simplified models.
  • 31.03.15   10:20

  • Stanislav Hencl
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • Jacobians of Sobolev homeomorphisms
  • Abstract:
  • In models of nonlinear elasticity people always assume that the deformation is orientation preserving, i.e. the Jacobian does not change sign. We show that for homeomorphisms in dimension n=3 we can assume this without loss of generality, i.e. each homeomorphism satisfies either J_fgeq 0 a.e. or J_fleq 0 a.e. This is a joint work with J. Maly.
  •                    09:00

  • Manoj K. Yadav
  • (Faculty of Civil Engeneering, Czech Technical University in Prague)
  • On a result of N. Masmoudi in homogenization
  • Abstract:
  • We will continue referring on the paper of Nader Masmoudi: Homogenization of the compressible Navier-Stokes equations in a porous medium.
  • 24.03.15   10:20

  • Ondrej Kreml
  • (Institute of Mathematics, Academy of Sciences of the Czech Republic)
  • Uniqueness of rarefaction waves in compressible Euler systems
  • Abstract:
  • We consider two systems of partial differential equations; the compressible isentropic Euler system and the complete Euler system describing the time evolution of an inviscid nonisothermal gas. In both cases we show that the rarefaction wave solutions of the 1D Riemann problem are unique in the class of all bounded weak solutions to the associated multi-D problem. This may be seen as a counterpart of the non-uniqueness results of physically admissible solutions emanating from 1D shock waves constructed recently by the method of convex integration.
  •                    09:00

  • Manoj K. Yadav
  • (Faculty of Civil Engeneering, Czech Technical University in Prague)
  • On a result of N. Masmoudi in homogenization
  • Abstract:
  • We will refer the paper of N. Masmoudi - Homogenization of the compressible Navier-Stokes equations in a porous medium.
  • 17.03.15   10:20

  • Miroslav Bulicek
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • Nonlinear elliptic equations beyond the natural duality pairing
  • Abstract:
  • Many real-world problems are described by nonlinear partial differential equations. A promiment example of such equations is nonlinear (quasilinear) elliptic system with given right hand side in divergence form div f data. In case data are good enough (i.e., belong to L^2), one can solve such a problem by using the monotone operator therory, however in case data are worse no existence theory was available except the case when the operator is linear, e.g. the Laplace operator. For this particular case one can however establish the existence of a solution whose gradient belongs to L^q whenever f belongs to L^q as well. From this point of view it would be nice to have such a theory also for general operators. However, it cannot be the case as indicated by many counterexamples. Nevertheless, we show that such a theory can be built for operators having asymptotically the Uhlenbeck structrure, which is a natural class of operators in the theory of PDE.
  •                    09:00

  • Bernard Ducomet
  • (CEA, DAM, DIF)
  • The P1 approximation for viscous barotropic and radiating flows
  • Abstract:
  • We consider a simplified model of radiative hydrodynamics consisting in coupling the barotropic Navier-Stokes system to the "P1" approximation of the radiative transfer equation. In the critical regularity setting, we prove global existence for data near a stable equilibrium. We also discuss some pertinent asymptotics (low Mach and diffusion limits). These results have been obtained in collaboration with Raphael Danchin (LAMA, Universite Paris Est).
  • 10.03.15   09:00

  • Martin Kalousek
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • Introductory lecture to homogenization problems
  • 03.03.15   10:20

  • Petr Kaplicky
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • BMO estimates for some elliptic problems
  • Abstract:
  • We will discuss BMO estimates for a weak solutions of the inhomogeneous p-Laplace system given by $-div(|nabla u|^{p-2} nabla u) = div f$. We show that $f in BMO$ implies $|nabla u|^{p-2} nabla u in BMO$, which is the limiting case of the nonlinear Calderon-Zygmund theory. This extends the work of DiBenedetto and Manfredi (1993), which was restricted to the super-quadratic case $pgeq 2$, to the full case $p in (1,infty)$ and even more general growth. We also briefly mention an application of the method to steady planar flows of generalized Newtonian fluids.
  •                    09:00

  • Marius Tucsnak
  • (Université de Lorraine)
  • Estimatability and observers for a model of population dynamics with diffusion and age dependence
  • Abstract:
  • We consider a model of McKendrick type for population dynamics
    with age dependence and diffusion. We prove that, Using various
    observation operators, we obtain an infinite system which is
    estimatable (detectable). This information is used to construct
    observers able to reconstruct population dynamics in the whole spatial
    domain from measures in an arbitrarily small spatial and age domain.
  • 27.01.15   09:00

  • Peter Takac
  • (Universitat Rostock)
  • Space-Time Analyticity of Solutions to Linear and Semilinear Parabolic Equations in the Whole Space with Applications to Two Volatility Models in Mathematical Finance
  • Abstract:
  • We begin by a brief presentation of a well-known mathematical model for the European option pricing in a market with stochastic volatility (J.-P. Fouque, G. Papanicolaou, and K. R. Sircar). We will use only intuitive probabilistic arguments in order to explain the main ideas of the arbitrage-free option pricing introduced in the works of F. Black and M. Scholes and (independently) R. C. Merton in 1973. These probabilistic arguments, combined with Ito's formula, yield an interesting parabolic partial differential equation (or a system of weakly coupled equations) for the option price(s). We briefly explain why, from the point of view of Mathematical Finance (complete markets), it is of interest to study a parabolic system of precisely N coupled scalar parabolic equations on an N-dimensional Euclidean space (or a cone).
    Then we proceed to the more technical, mathematical part of our lectures. We treat first the classical Black-Scholes-type model, i.e., a linear parabolic system for which we prove a theorem on the analytic smoothing property of a uniformly parabolic system with analytic coefficients, with only L^2-Lebesgue integrable initial data over R^N. The main difficulty in our approach will be to establish suitable a priori L^2-type estimates for the holomorphic extension of the weak solution to a complex strip around R^N. Such estimates are allowed to depend solely on the L^2-norm of the initial data over R^N. We will (have to) investigate how the width of the complex strip around R^N increases with (the real part of) time. We use the Hardy spaces of holomorphic functions to do this. The analyticity in time is obtained in a much more familiar way that takes advantage of the theory of holomorphic semigroups. This part is based on the author's recent work (2012).
    In the second lecture we treat a semilinear generalization of the linear model with fairly general analytic nonlinearities. We obtain the necessary uniform bounds (local in time and global in space) on the weak solution to the nonlinear system by an abstract (real) interpolation method combined with maximal regularity. To prove the time-analytic smoothing property, we adapt several ideas of S. Angenent (1990) to our functional L^p-setting in time. However, the global uniform bounds in space are obtained only for those initial data that themselves obey these bounds. This process takes place in an interpolation trace space of functions with fractional smoothness, i.e., in a Besov space. As this approach yields also a priori bounds for the weak solution, local in time and global in space, it is then easy to perform a linearization procedure to obtain a linear equation for the difference of two weak solutions with different initial data. The L^2-norm of this difference is thus controlled by the L^2-norm of the difference of the initial data on R^N. This result provides the same type of a priori estimate as for the linear system and is obtained in an analogous way. This part is based on the author's ongoing work with a Ph.D. student (2015).
  • 13.01.15   10:20

  • Emil Wiedemann
  • (Universitat Bonn)
  • Measure-Valued Solutions of the Euler Equations
  • Abstract:
  • Measure-valued solutions of the incompressible Euler equations were first considered by DiPerna and Majda to describe effects of oscillation and concentration in ideal fluids. Although measure-valued solutions appear a priori as much weaker objects than distributional solutions, we have been able to show that both notions are in a sense equivalent. An important open question concerns the relation between weak and measure-valued solutions for compressible Euler models. Joint work with L. Székelyhidi, Jr.
  •                    09:00

  • Peter Takac
  • (Institut fur Mathematik, Universitat Rostock)
  • Space-Time Analyticity of Solutions to Linear and Semilinear Parabolic Equations in the Whole Space with Applications to Two Volatility Models in Mathematical Finance
  • Abstract:
  • We begin by a brief presentation of a well-known mathematical model for the European option pricing in a market with stochastic volatility (J.-P. Fouque, G. Papanicolaou, and K. R. Sircar). We will use only intuitive probabilistic arguments in order to explain the main ideas of the arbitrage-free option pricing introduced in the works of F. Black and M. Scholes and (independently) R. C. Merton in 1973. These probabilistic arguments, combined with Ito's formula, yield an interesting parabolic partial di erential equation (or a system of weakly coupled equations) for the option price(s). We brie y explain why, from the point of view of Mathematical Finance (complete markets), it is of interest to study a parabolic system of precisely N coupled scalar parabolic equations on an N-dimensional Euclidean space (or a cone).
    Then we proceed to the more technical, mathematical part of our lectures. We treat first the classical Black-Scholes-type model, i.e., a linear parabolic system for which we prove a theorem on the analytic smoothing property of a uniformly parabolic system with analytic coefficients, with only L^2-Lebesgue integrable initial data over R^N. The main difficulty in our approach will be to establish suitable a priori L^2-type estimates for the holomorphic extension of the weak solution to a complex strip around R^N. Such estimates are allowed to depend solely on the L^2-norm of the initial data over R^N. We will (have to) investigate how the width of the complex strip around R^N increases with (the real part of) time. We use the Hardy spaces of holomorphic functions to do this. The analyticity in time is obtained in a much more familiar way that takes advantage of the theory of holomorphic semigroups. This part is based on the author's recent work (2012).
    In the second lecture we treat a semilinear generalization of the linear model with fairly general analytic nonlinearities. We obtain the necessary uniform bounds (local in time and global in space) on the weak solution to the nonlinear system by an abstract (real) interpolation method combined with maximal regularity. To prove the time-analytic smoothing property, we adapt several ideas of S. Angenent (1990) to our functional L^p-setting in time. However, the global uniform bounds in space are obtained only for those initial data that themselves obey these bounds. This process takes place in an interpolation trace space of functions with fractional smoothness, i.e., in a Besov space. As this approach yields also a priori bounds for the weak solution, local in time and global in space, it is then easy to perform a linearization procedure to obtain a linear equation for the difference of two weak solutions with different initial data. The L^2-norm of this difference is thus controlled by the L^2-norm of the difference of the initial data on R^N. This result provides the same type of a priori estimate as for the linear system and is obtained in an analogous way. This part is based on the author's ongoing work with a Ph.D. student (2015).
  • 06.01.15   09:00

  • Agnieszka Swierczewska-Gwiazda
  • (University of Warsaw)
  • Transport equation with integral terms
  • Abstract:
  • We will consider the transport equation with Sobolev coefficients and a right-hand side in form of an integral operator. Such problem cannot be solved directly by means of renormalization techniques and the essential step is formulating the problem in terms of regular Lagrangian flows. The talk is based on common results with Camillo De Lellis and Piotr Gwiazda.
  • 16.12.14   10:20

  • Piotr Gwiazda
  • (University of Warsaw)
  • Analysis of viscosity models for concentrated polymers
  • Abstract:
  • There are numerous studies on dilute polymers including FENE model, Doi model and others. The short overview of them will be presented. The main attention will be however directed to the description of the flow of concentrated polymers where the length of polymer chains affects the viscosity of the fluid.
    The talk is based on the common result with Miroslav Bulicek, Endre Suli and Agnieszka Swierczewska-Gwiazda.
  •                    09:00

  • Sebastian Schwarzacher
  • On the time derivative of degenerated parabolic PDEs
  • Abstract:
  • In my talk I will discuss regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system as well as models for non Newtonien fluids of powerlaw or Carreau type. By the use of special weak different quotients, adapted to the variational structure it is possible to get fractional derivatives of u_t in time and space direction.
  • 09.12.14   09:00

  • Milan Pokorny
  • (MFF UK)
  • On the existence of weak solutions to the equations of steady flow
  • Abstract:
  • We study a system of partial differential equations describing the steady flow of a heat conducting incompressible fluid in a bounded three dimensional domain, where the right-hand side of the momentum equation includes the buoyancy force. In the present work we prove the existence of a weak solution under both the smallness and a sign condition on physical parameters alpha_0 and alpha _1 which appear on the right hand side. It is a joint work with J. Naumann and J. Wolf (Berlin).
  • 02.12.14   09:00

  • Lu Yong
  • (MFF UK)
  • On PDE analysis of flows of quasi-incompressible fluids
  • Abstract:
  • We study mathematical properties of quasi-compressible fluids. These are mixtures in which the density depends on the concentration of one of their components. Assuming that the mixture meets mass and volume additivity constraints, this density-concentration relationship is given explicitly. We show that such a constrained mixture can be written in the form similar to compressible Navier-Stokes equations with a singular relation between the pressure and the density. This feature automatically leads to the density bounded from below and above. After addressing the choice of thermodynamically compatible boundary conditions, we establish the large data existence of weak solution to the relevant initial and boundary value problem. We then investigate one possible limit from a quasi compressible to incompressible regime.
  • 25.11.14   09:00

  • Tomas Barta
  • (MFF UK)
  • Lojasiewicz-type inequalities and convergence of solutions to equilibrium
  • Abstract:
  • If a global solution to an evolution equation is relatively compact, then it has an accumulation point. We provide some well-known (Lojasiewicz, resp. Kurdyka-Lojasiewicz) estimates and some new conditions that imply convergence of the solution. The abstract result will be applied to show convergence of a wave equation with a general damping function.
  • 18.11.14   09:00

  • Aneta Wroblewska-Kaminska
  • (IMPAN)
  • Non-Newtonian fluids and abstract problems: application of Orlicz spaces in the theory of nonlinear PDE
  • Abstract:
  • We are interested in the existence of solutions to strongly nonlinear partial differential equations. We concentrate mainly on problems which come from dynamics of non-Newtonian fluids of a nonstandard rheology, more general then of power-law type, and abstract theory of elliptic and parabolic equations. In considered problems the nonlinear highest order term (stress tensor) is monotone and its behaviour - coercivity/growth condition - is given with help of some general convex function. In our research we would like to cover both cases: sub- and super-linear growth of nonlinearity (shear thickening and shear tinning fluids) as well its anisotropic and non-homogenous behaviour. Such a formulation requires a general framework for the function space setting, therefore we work with non-reflexive and non-separable anisotropic Orlicz and Musielak-Orlicz spaces. Within the presentation we would like to emphasise problems we have met during our studies, their reasons and methods which allow us to achieve existence results.
  • 11.11.14   09:00

  • Minsuk Yang
  • (KIAS)
  • Existence and uniqueness for the magnetohydrodynamic equations in the Besov space
  • Abstract:
  • We consider the Cauchy problem of the incompressible magnetohydrodynamic equations with no magnetic diffusion term in three spatial dimension. This model appears in astrophysics. We consider existence and uniqueness when the initial data are in a certain homogeneous Besov space. For this purpose, we shall review some of the basic facts in harmonic analysis. This is a joint work with Hi Jun Choe.
  • 04.11.14   10:20

  • Filippo Dell'Oro
  • (IM CAS)
  • Asymptotic analysis of thermoelastic systems with Gurtin-Pipkin thermal law
  • Abstract:
  • We provide a comprehensive stability analysis of the thermoelastic Timoshenko and Bresse systems. In particular, assuming a temperature evolution of Gurtin Pipkin type, we establish a necessary and sufficient condition for exponential stability in terms of the structural parameters of the problem. As a byproduct, a complete characterization of the longtime behavior of Timoshenko and Bresse systems with Fourier, Maxwell-Cattaneo and Coleman-Gurtin thermal laws is obtained.
  •                    09:00

  • Prof. Dr. Hans Knuepfer
  • (Uni Heidelberg)
  • Multiple-droplet phases for a charged liquid in a neutralizing background
  • Abstract:
  • We consider a macroscopic limit for the Ohta-Kawasaki energy. This model has been used to described to describe phase separation for diblock-copolymers. We first investigate existence and shape of minimizers of the energy with prescribed volume (of the one phase) in the full space setting. We then consider situation of periodic configurations with prescribed density of the minority phase. We show that in a certain regime, the energy Gamma-converges to a homogenized problem. This is joint work with C. Muratov and M. Novaga.
  • 21.10.14   09:00

  • Elisabetta Chiodaroli
  • (École polytechnique fédérale de Lausanne)
  • An overview on some recent results for the Euler system of isentropic gas dynamics
  • Abstract:
  • This talk is concerned with the well-posedness problem for the isentropic
    compressible Euler equations of gas dynamics, the oldest but yet most prominent
    paradigm for hyperbolic systems of conservation laws. The results We present are in the line with the program of investigating the efficiency of different selection criteria proposed in the literature in order to weed out non-physical solutions to more-dimensional systems of conservation laws and they build upon the method of convex integration developed by De Lellis-Székelyhidi for the incompressible Euler equations. Inspired by these interesting question, we first recall some counterexamples to uniqueness of entropy solutions to the Cauchy problem for the multi-dimensional compressible Euler equations: in our construction the entropy condition is not suffficient as a selection criterion for unique solutions. We will then devote our attention to the role of the maximal dissipation criterion proposed by Dafermos. Specifically, we will illustrate how some non-standard (i.e. constructed via con-vex integration methods) solutions to the Riemann problem for the isentropic Euler system in 2 space dimensions have greater energy dissipation rate than the classical self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour in general the self-similar solutions. (This is joint work with Camillo De Lellis and Ondrej Kreml)
  • 15.10.14   09:00

  • Martin Michalek
  • (IM CAS)
  • Compressible Navier-Stokes with Entropy Transport (stability result)
  • Abstract:
  • We show a stability result of the compressible Navier-Stokes system with transport equation for entropy. The proof comes as an outcome of the isentropic case and additional properties of the effective viscous flux. We deal with adiabatic index gamma>3/2 in the pressure term; the crucial renormalization techniques are therefore restricted.
  • 07.10.14   09:00

  • Martina Hofmanova
  • A regularity result for quasilinear parabolic SPDE's
  • Abstract:
  • We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is H"older continuous in time and possesses spatial regularity that is only limited by the regularity of the given data.
    Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problém whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs. It is a joint work with Arnaud Debussche and Sylvain de Moor.
  • 17.09.14   10:30

  • Dominic Breit
  • Stochastic Navier-Stokes equations for compressible fluids
  • Abstract:
  • We study the Navier-Stokes equations governing the motion of isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density, which is affine linear in momentum and satisfies suitable growth assumptions with respect to density, and establish existence of the so-called finite energy weak martingale solution under the condition that the adiabatic constant satisfies $gamma>3/2$. The proof is based on a four layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.
  •                    09:00

  • Sarka Necasova
  • (Institute of Mathematics, CAS)
  • Low Mach number limit and diffusion limit in a model of radiative flow
  • Abstract:
  • We consider an asymptotic regime for a simplified model of compressible Navier-Stokes-Fourier system coupled to the radiation, when hydrodynamical flow is driven to incompressibility through the low Mach number limit. We prove a global in-time existence for the primitive problem in the framework of weak solutions and for the incompressible target system and we study the convergence of the primitive system toward its incompressible limit. Moreover,we investigate the cases when the radiative intensity is driven either to equilibrium or to non-equilibrium diffusion limit, depending the scaling performed, and we study the convergence of the system toward the aforementioned limits.
  • 20.05.14   09:00

  • Petr Kucera
  • (Czech Technical University)
  • Strong solutions of the Navier-Stokes equations with Navier's boundary conditions
  • Abstract:
  • In this contribution we deal with a system of the Navier-Stokes equations with Navier's boundary condition or with Navier-type boundary conditions. We study perturbations of initial conditions of strong solutions of our system. We prove that if these perturbations are sufficiently small in L^3 norm then corresponding solutions are strong too.
  • 13.05.14   09:00

  • Antonin Novotny
  • (IMATH, Universite du Sud Toulon-Var)
  • Error estimates for the compressible Navier-Stokes equations
  • 29.04.14   09:00

  • Milan Pokorny
  • (MFF UK)
  • A Linearized Model for Compressible Flow past a Rotating Obstacle: Analysis via Modified Bochner-Riesz Multipliers
  • Abstract:
  • We consider the flow of a compressible Newtonian fluid around or past a rotating rigid obstacle in R^3. After a coordinate transform to get a problem in a time-independent domain we assume the new system to be stationary. We linearize it and use Fourier transform to prove the existence of a unique solution in L^q-spaces. However, in contrast to the incompressible case with multipliers based on the heat kernel the new multiplier functions are related to Bochner-Riesz multipliers and require the restriction 6>q>6/5.
  • 22.04.14   09:00

  • Zdenek Skalak
  • (Czech Technical University in Prague)
  • Several notes on the conditional regularity for the solutions of the Navier-Stokes equations
  • Abstract:
  • We study regularity criteria for the nonstationary solutions of the Navier-Stokes equations in the whole three-dimensional space based on the velocity gradient. We pay a special attention to the criteria based on the regularity of the gradient of one velocity component. We use the Lebesgue and Besov spaces for the presentation of our results.
  • 15.04.14   09:00

  • Martin Kalousek
  • (MFF UK)
  • Homogenization of a non-Newtonian flow through a porous medium
  • Abstract:
  • We consider the stationary incompressible viscous non-Newtonian flow through a porous medium. We assume that viscosity is a nonlinear function of the symmetric velocity gradient, i.e. this nonlinear function is a generalization of the power-law case. We provide a mathematical derivation of the law governing a polymer flow through a porous medium using homogenization. The crucial mathematical tool that we use is two-scale convergence, here adopted for Orlicz setting.
  • 08.04.14   09:00

  • Josef Zabensky
  • (MFF UK)
  • On a generalization of the Darcy-Forchheimer equation
  • Abstract:
  • We study mathematical properties of steady flows described by the system of equations generalizing the classical porous media models of Darcy's and Forchheimer's. The considered generalizations are outlined by implicit relations between the drag force and the velocity, that are in addition parametrized by the pressure. We analyze such drag force--velocity relations which are described through a maximal monotone graph varying continuously with the pressure. Large-data existence of a solution to this system is established, whereupon we show that under certain assumptions on data, the pressure satisfies a maximum or minimum principle, even if the drag coefficient depends on the pressure exponentially.
  • 01.04.14   09:00

  • Hana Mizerov
  • (Univ. of Mainz)
  • Existence of the weak solution for the Peterlin viscoelastic model
  • Abstract:
  • We consider the viscoelastic model describing the behavior of some polymeric fluids. The polymer molecules are treated as two beads connected by a nonlinear spring. The Peterlin approximation of the spring force is used to derive the equation for the conformation tensor. The aim of this talk is to present the existence results for this model. Moreover, assuming more regular initial data we get unique solutions. This reserach has been done in the collaboration with Maria Lukacova and Sarka Necasova. It has been supported by the German Research Foundation under the IRTG "Mathematical Fluid Dynamics".
  • 18.03.14   09:00

  • Vaclav Macha
  • (Institute of Mathematics, CAS)
  • Self-propelled motion in a viscous compressible fluid
  • Abstract:
  • We focus on an existence of a weak solution to a system describing a self-propelled motion of a single deformable body in a viscous compressible fluid which occupies a bounded domain in the 3 dimensional Euclidean space. The considered governing system for the fluid is the isentropic compressible Navier-Stokes equation. We present a proof an existence of a weak solution up to a collision.
  • 04.03.14   09:00

  • Ondrej Kreml
  • (Institute of Mathematics, CAS)
  • On bounded solutions to the compressible isentropic Euler system
  • Abstract:
  • We analyze the Riemann problem for the compressible isentropic Euler system in the whole space $mathbb{R}^2$. Using the tools developed by De Lellis and Sz'ekelyhidi for the incompressible Euler system we show that for every Riemann initial data yielding the self-similar solution in the form of two admissible shocks there exist in fact infinitely many admissible bounded weak solutions. Moreover for some of these initial data such solutions dissipate more total energy than the self-similar solution which might be looked at as a natural candidate for the "physical" solution. Finally, we show using the relative entropy inequality that self-similar solutions consisting only of rarefaction waves are unique in the class of bounded admissible weak solutions.
  • 25.02.14   09:00

  • Antonin Novotny
  • (IMATH, University of Toulon)
  • Some topics in the mathematical thermodynamics of compressible fluids II.
  • Abstract:
  • We will talk about several issues related to the notions of weak solutions, dissipative solutions and stability properties to the compressible Navier-Stokes system and its approximations.
  • 18.02.14   09:00

  • Bum Ja Jim
  • (Department of Mathematics Education, Mokpo National University, Muan-gun 534-729, South Korea)
  • ON THE REGULARITY OF WEAK SOLUTIONS TO THE MOTION OF THE DEGENERATE POWER-LAW FLUIDS
  • Abstract:
  • Let Ω be a bounded domain in Rn; n = 2; 3. We consider the steady
    and unsteady motion of a fluid described by the systems
    (u · ∇)u − divS(Du) + ∇p = f; divu = 0 in Ω; u|@Ω = 0 (0.1)
    and
    ut+(u·∇)u−divS(Du)+∇p = 0; divu = 0 in Ω×(0; T); u|@Ω = 0; u|t=0 = a; (0.2)
    respectively, where
    S(Du) = |Du|^(q-2)Du. (0.3)
    When q = 2, the system becomes (incompressible)Navier-Stokes equations, and the well known theories on the linear (partial differential)operator could be applied to study the regularity of weak solutions.
    When q ̸= 2, the structure of stress tensor is no more linear, whence we cannot apply any known linear operator theory. To study regularity of weak solutions it is natural to use variational approach such as difference quotient scheme.
  • 11.02.14   10:20

  • Camillo De Lellis
  • (University of Zurich)
  • From Nash to Onsager: funny coincidences across differential geometry and the theory of turbulence
  • Abstract:
  • The incompressible Euler equations were derived more than 250 years ago by Euler to describe the motion of an inviscid incompressible fluid. It is known since the pioneering works of Scheffer and Shnirelman that there are nontrivial distributional solutions to these equations which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force. A celebrated theorem by Nash and Kuiper shows the existence of C^1 isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the threedimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it. In a first joint work with Laszlo Szekelyhidi we pointed out that these two counterintuitive facts share many similarities. This has become even more apparent in some recent results of ours, which prove the existence of Hoelder continuous solutions that dissipate the kinetic energy. Our theorem might be regarded as a first step towards a conjecture of Lars Onsager, which in his 1949 paper about the theory of turbulence asserted the existence of such solutions for any Hoelder exponent up to 1/3. Currently the best result in this direction, 1/5, has been reached by Phil Isett.
  •                    09:00

  • Antonin Novotny
  • (IMATH, University of Toulon)
  • Some topics in the mathematical thermodynamics of compressible fluids I.
  • Abstract:
  • We will talk about several issues related to the notions of weak solutions, dissipative solutions and stability properties to the compressible Navier-Stokes system and its approximations.
  • 04.02.14   10:20

  • Hi Jun Choe
  • (Yonsei Univ., Seoul, Korea)
  • Compressible Navier-Stokes Limit of Binary Mixture of Gas Particles
  • Abstract:
  • In this talk we study compressible Navier-Stokes limit of binary mixture of gas particles in which a species is dense and the other is sparse. Their collisions are decided by Grad's hard
    potentials.
    When Knudsen number of dense species of Boltzmann system goes to zero, we show that the hydrodynamic variables satisfy compressible Navier-Stokes type equations.
    It turns out that the macro fluid variables corresponding to the dense species satisfy the standard compressible Navier-Stokes equations. But the fluid equations for sparse species contain influence terms of dense species.
    Like single species gas, we employed Enskog-Chapman and moment methods up to the first order.
  •                    09:00

  • Aneta Wroblewska-Kaminska
  • (Institute of Mathematics, Polish Academy of Sciences)
  • The Oberbeck-Boussinesq approximation in R3 as a limit of compressible Naver-Stokes-Fourier with low Mach number
  • Abstract:
  • We will present the asymptotic analysis of solutions to the compressible Navier-Stokes-Fourier system, when the Mach number is small proportional to $varepsilon$, Froud number is proportional to $sqrt varepsilon$ and  $varepsilon rightarrow 0$ and the domain containing the fluid varies with changing parameter . In particular, the fluid is driven by gravitation generated by object(s) placed in fluid of diameter converging to zero. As  $varepsilon rightarrow 0$, we will show that the fluid velocity converges to a solenoidal vector eld satisfying the Oberbeck-Boussinesq approximation on R3 space with concentric gravitation force. The proof is based on spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.
  • 14.01.14   09:00

  • László Székelyhidi
  • (Universität Leipzig)
  • Weak solutions of the Euler equations: non-uniqueness and dissipation
  • Abstract:
  • There are two aspects of weak solutions of the incompressible Euler equations which are strikingly different to the behaviour of classical solutions. Weak solutions are not unique in general and do not have to conserve the energy. Although the relationship between these two aspects is not clear, both seem to be in vague analogy with Gromov’s h-principle. In the talk I will explore this analogy in light of recent results concerning both the non-uniqueness, the search for selection criteria, as well as the dissipation anomaly and the conjecture of Onsager.
  • 07.01.14   10:20

  • Peter Takác
  • (Universität Rostock)
  • Travelling waves in a Fisher-Kolmogorov-type model with degenerate diffusion and nonsmooth reaction
  • Abstract:
  • We will discuss the existence and uniqueness of monotone travelling waves connecting the equilibrium states +-1.
    They can either only approach these equilibria at +-infinity, or else attain them at finite points, depending on the interaction between the degenerate / singular diffusion and the nonsmooth reaction function. Then we discuss the approach to such travelling waves by solutions with rather general initial data that are sqeezed between two travelling waves (that are each other's shift).
  •                    09:00

  • Xavier Blanc
  • (Université Paris Diderot (Paris 7))
  • Global weak solutions to the 1D compressible Euler equations with radiation
  • Abstract:
  • We consider the Cauchy problem for the equations of one-dimensional motion of a compressible inviscid gas coupled with radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients and the data, we prove that the problem admits a weak solution. More precisely, we show that a sequence of approximate solutions constructed by a generalized Glimm's scheme admits a subsequence converging to an entropic solution of the problem.
  • 03.12.13   09:00

  • Davide Catania
  • (Universitŕ di Brescia)
  • Analysis of the Oberbeck-Boussinesq Equations with some anisotropic filters.
  • Abstract:
  • We introduce and analyze a new family of large scale methods to simulate mixing and to perform numerical simulations of the Oberbeck-Boussinesq equations, with possible applications to modeling of certain geophysical flows as oceanic models or volcanic plumes, when direct numerical simulation is not possible.
    We consider approximate models, based on suitable anisotropic filters, for the Boussinesq equations in bounded domains, where part of the boundary satisfies periodic conditions, and the remaining part homogeneous Dirichlet conditions.
    We discuss the existence and well-posedness of the solutions, with particular emphasis on the role of the energy. We can show that all models possess suitable regular weak solutions that satisfy the energy inequality for the model, but only when all the equations are filtered, we can prove that the solution is unique and, moreover, satisfies an energy identity.
  • 19.11.13   09:00

  • Marius Tucsnak
  • (Université de Lorraine)
  • Controllability of a simplified fluid-structure interaction model - ERC MathEF Lecture
  • Abstract:
  • We consider a system formed by a viscous 1D Burger equations in a domain with a free boundary, coupled with the Newton laws for the motion of the interface. This system can be seen as a "toy model" for the equations modeling the motion of a rigid body in a viscous fluid. We prove controllability results with inputs acting either on the exterior or the free boundaries.
  • 14.11.13   09:30

  • Marius Tucsnak
  • (Université de Lorraine)
  • Efficient swimminng at low Reynolds number - ERC MathEF Lecture
  • Abstract:
  • We show that swimming at low Reynolds number can be interpreted as a finite dimensional control problem. For simple shaped micro-swimmers, we analyze controllability properties and we tackle associated optimal control problems.
  • 12.11.13   09:00

  • Donatella Donatelli
  • (University of L'Aquila)
  • Zero electron mass limit of a hydrodynamic model for charge-carrier transport
  • Abstract:
  • We are concerned with the rigorous analysis of the zero electron mass limit of hydrodynamic model for charge-carrier transport. The model is given by the full Navier-Stokes-Poisson system. This system has been introduced in the literature by Anile and Pennisi (see [1]) in order to describe a hydrodynamic model for charge-carrier transport in semiconductor devices. The purpose of our work is to prove rigorously zero electron mass limit in the framework of general ill prepared initial data. In this situation the velocity field and the electronic fields develop fast oscillations in time. The main idea we are using is a combination of formal asymptotic expansion and rigorous uniform estimates on the error terms. Finally we prove the strong convergence of the full Navier Stokes Poisson system towards the strong solutions of the incompressible Navier Stokes equations plus a term of fast singular oscillating gradient vector fields (see [2]).

    [1] A. Anile, S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Physical Review B, 46, no. 20 (1992), 13186--13193.
    [2] L. Chen, D. Donatelli, P. Marcati, Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport, SIAM J. Math. Analysis, 45, no.3 (2013), 915-933
  • 05.11.13   10:20

  • David Krejcirik
  • (UJF AVCR)
  • The effective Hamiltonian in curved quantum waveguides and when it does not work
  • Abstract:
  • The Dirichlet Laplacian in a curved three-dimensional tube
    built along a spatial (bounded or unbounded) curve is investigated
    in the limit when the uniform cross-section diminishes.
    Both deformations due to bending and twisting are considered.
    We show that the Laplacian converges in a norm resolvent sense
    to the well known one-dimensional Schroedinger operator
    whose potential is expressed in terms of the curvature
    of the reference curve, the twisting angle
    and a constant measuring the asymmetry of the cross-section.

    Contrary to previous results, we allow reference curves
    to have non-continuous and possibly vanishing curvature.
    For such curves, the distinguished Frenet frame need not exist
    and, moreover, the known approaches to establish the result
    do not work. We ask the question under which minimal
    regularity assumptions the effective one-dimensional
    approximation holds.

    Our main ideas how to establish the norm-resolvent convergence
    under the minimal regularity assumptions are to use an alternative
    frame defined by a parallel transport along the curve and a refined
    smoothing of the curvature via the Steklov approximation.
    On the negative side, we construct an explicit waveguide
    for which the usefulness of the spectral information provided
    by the effective Hamiltonian is rather doubtful.
  •                    09:00

  • Sergei Kuksin
  • ((ERC Mathef))
  • KdV equation under periodic boundary conditions and its perturbations
  • Abstract:
  • I will discuss properties of the KdV equation under periodic boundary conditions, relevant to study perturbations of the equation. Next I will review what is known now about long-time behaviour of solutions for the perturbed equations and discuss open problems.
  • 29.10.13   09:00

  • Trygve Karper
  • (University of Maryland)
  • Convergence of numerical methods for viscous compressible flow
  • Abstract:
  • In this mini-course, the goal is to introduce the audience
    to numerical analysis of methods for compressible gas flow.
    The course can be roughly divided in two parts.

    In the first part, we will review the basic methodology
    used to develop numerical methods for compressible flow.
    Moreover, we will review some of their basic properties such
    as stability, accuracy, and structure preservation.

    In the second part, we will review the
    recently developed convergence theory for numerical
    approximations of the compressible Navier-Stokes equations.
    In particular, we will demonstrate that sequences of numerical
    solutions converge weakly to a weak solution of the continuous
    problem.
  • 24.10.13   10:20

  • Antonín Novotný
  • (IMATH, Universite du Sud Toulon-Var)
  • Relative energy method for the compressible Navier-Stokes equations.
  • Abstract:
  • In this mini-course, we shall derive the relative entropy inequality for the compressible Navier-Stokes equations and investigate some of its applications in the thermodynamics of compressible fluids.
  •                    09:00

  • Trygve Karper
  • (University of Maryland)
  • Convergence of numerical methods for viscous compressible flow
  • Abstract:
  • In this mini-course, the goal is to introduce the audience
    to numerical analysis of methods for compressible gas flow.
    The course can be roughly divided in two parts.

    In the first part, we will review the basic methodology
    used to develop numerical methods for compressible flow.
    Moreover, we will review some of their basic properties such
    as stability, accuracy, and structure preservation.

    In the second part, we will review the
    recently developed convergence theory for numerical
    approximations of the compressible Navier-Stokes equations.
    In particular, we will demonstrate that sequences of numerical
    solutions converge weakly to a weak solution of the continuous
    problem.
  • 22.10.13   10:20

  • Antonín Novotný
  • (IMATH, Universite du Sud Toulon-Var)
  • Relative energy method for the compressible Navier-Stokes equations.
  • Abstract:
  • In this mini-course, we shall derive the relative entropy inequality for the compressible Navier-Stokes equations and investigate some of its applications in the thermodynamics of compressible fluids.
  •                    09:00

  • Jiří Neustupa
  • (Institute of Mathematics, CAS)
  • Regularity of a suitable weak solution to the Navier-Stokes equations via a spectral projection of vorticity.
  • Abstract:
  • It is well known that certain rate of integrability of vorticity (or its two components), corresponding to a weak solution of the Navier-Stokes equations, implies regularity of the weak solution. We show that sufficient conditions for regularity can also be imposed only on certain spectral projection of vorticity or, particularly, on only one its component.
  • 15.10.13   09:00

  • Xian Liao
  • (Mathematical Institute, Charles University in Prague)
  • On the well-posedness of the low Mach-number limit system in the optimal Besov spaces
  • Abstract:
  • This talk is contributed to the well-posedness issue of the low Mach-number limit system in the optimal Besov spaces. The system derives formally from the full Navier-Stokes-Fourier system when the Mach number tends to vanish. Since the thermal conduction is also considered here, a diffusion effect (relating nonlinearly the velocity and the density) will occur in the limit condition, unlike in the case without thermal diffusion where only the incompressibility condition will come out. A priori estimates and estimates for nonlinearities (e.g. products, commutators) are based on the Littlewood-Paley theory and paradifferential calculus.
  • 08.10.13   09:00

  • Cristian Cazacu
  • (Simion Stoilow Institute)
  • The influence of the Hardy-type inequalities to some singular PDEs
  • Abstract:
  • In this talk we discuss qualitative Hardy-type inequalities for Schrödinger operators of the form $A_lambda:=-Delta -lambda V$, $lambda>0$, where $V$ is a positive potential with quadratic singularities located either in the interior or on the boundary of an open smooth domain $Omegasubset mathbb{R}^N$, $Ngeq 1$.  We discuss the criticality of $A_lambda$ in terms of the number and location of the singular poles. In addition, due to the presence of the singular potential $V$, standard elliptic regularity of the Dirichlet problem associated to $A_lambda$ fails. In particular, we emphasize how this impediment affects  the controllability aspects of the wave equation with the singular potential $V=1/|x|^2$.
  • 01.10.13   09:00

  • Jorg Wolf
  • (Department of Mathematics Humboldt University Berlin)
  • Partial regularity of local suitable weak solutions to the system of generalized Newtonian uids with power low q > 2
  • Abstract:
  • In our present talk we going to extend the famous Ca farelli-Kohn-Nirenberg theorem for suitable weak solutions to the Navier-Stokes equations to local suitable weak solutions of a system of power law fluid.
  • 26.09.13   10:00

  • Elisabetta Chiodaroli
  • Non-standard solutions to the compressible Euler system
  • Abstract:
  • The deceivingly simple-looking compressible Euler equations of gas dynamics have a long history of important contributions over more than two centuries. If we allow for discontinuous solutions, uniqueness and stability are lost. In order to restore such properties further restrictions on weak solutions have been proposed in the form of entropy inequalities. In this talk, we will discuss some counterexamples to the well-posedness theory of entropy solutions to the multi-dimensional compressible Euler equations. All our methods are inspired by a new analysis of the incompressible Euler equations recently carried out by De Lellis and Szekelyhidi and based on a revisited “h-principle”ť.
  • 02.07.13   09:00

  • Vladimir Sverak
  • On invariant measures for Hamiltonian PDEs.
  • Abstract:
  • A natural first impression is that the dynamics in infinite dimensions should be more complicated than the dynamics in finite dimensions. This is true in some respects, but at the same time the infinite dimension can potentially simplify the "macroscopic" behavior, in a way similar to the simplifications conjectured by the ergodic hypothesis for many classical systems of Statistical Mechanics. Ergodicity-type assumptions are notoriously difficult to prove or disprove (and we have nothing new to say in this direction) , but there are many other interesting questions, some of which we will address. The talk will be based on joint works with Nathan Glatt-Holtz and Vlad Vicol, and with Geordie Richards and Ofer Zeitouni.
  • 28.06.13   10:00

  • Susan Friedlander
  • (University of Southern California)
  • Well/Ill-Posedness Results for the Magneto-Geostrophic Equations
  • Abstract:
  • We consider an active scalar equation with singular drift velocity that is motivated by a model for the geodynamo. We show that the
    non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In
    contrast, the critically diffusive equation is globally well-posed.
    The spatial anisotropy of the equation gives rise to some interesting novel effects which we discuss in the context of the equation with fractional diffusion.
    This work is joint with Vlad Vicol.
  • 27.06.13   11:15

  • Vladimir Sverak
  • On scale invariant solutions of the Navier-Stokes equations.
  • Abstract:
  • The study of the scale invariant solutions of the Navier-Stokes equation has revealed interesting possibilities for scenarios of ill-posedness for compactly supported finite-energy initial data just at the borderline of well-posedness results provided by standard perturbation theory. Sufficient conditions for ill-posedness of the Leray-Hopf solutions for such seemingly benign initial data involve only the spectrum of certain natural linear operators obtained from scale-invariant solutions. The validity of these sufficient condition can be investigated by relatively routine numerical computations. Based on joint work with Hao Jia.
  •                    10:00

  • Radek Erban
  • (University of Oxford)
  • Hybrid Modelling of Reaction, Diffusion and Taxis Processes in Biology
  • Abstract:
  • I will discuss methods for spatio-temporal modelling in cellular and
    molecular biology. Three classes of models will be considered:
    (i) microscopic (molecular-based, individual-based) models which are based on the simulation of trajectories of individual molecules and their localized interactions (for example, reactions);
    (ii) mesoscopic (lattice-based) models which divide the computational domain into a finite number of compartments and simulate the time evolution of the numbers of molecules in each compartment; and
    (iii) macroscopic (deterministic) models which are written in terms of reaction-diffusion-advection partial differential equations (PDEs) for spatially varying concentrations.
    In the first part of my talk, I will discuss connections between the modelling frameworks (i)-(iii). I will consider chemical reactions both at a surface and in the bulk. In the second part of my talk, I will present hybrid (multiscale) algorithms which use models with a different level of detail in different parts of the computational domain. The main goal of this multiscale methodology is to use a detailed modelling approach in localized regions of particular interest (in which accuracy and microscopic detail is important) and a less detailed model in other regions in which accuracy may be traded for simulation efficiency. I will also discuss hybrid modelling of chemotaxis where an individual-based model of cells is coupled with PDEs for extracellular chemical signals.
  • 05.06.13   10:00

  • Martin Kalousek
  • (Faculty of Mathematics and Physics, Charles University)
  • Homogenization of generalized Navier-Stokes equations
  • Abstract:
  • We consider the generalized Navier-Stokes equations describing the flow of a fluid which changes its rheological properties in the
    presence of electromagnetic field. The field is assumed to be periodic and its period is specified by a small parameter. The task of homogenization in this case is to find the effective form of the stress tensor which does not depend on the spatial variable.
  • 21.05.13   09:00

  • Martin Vaeth
  • (Free University Berlin)
  • Turing Instability for a Reaction-Diffusion System with Unilateral Obstacles
  • Abstract:
  • Starting from the concept of a stable equilibrium of two chemicals,
    Turing's fundamental idea of morphogenesis by diffusion is sketched,
    heuristically. Unfortunately, this idea requires an extreme asymmetry
    of diffusion speeds. It is explained, heuristically, why this idea
    works also without this requirement in the presence of a unilateral
    obstacle (e.g. a certain type of source for the inhibitor).
    Mathematically more precise and more quantitatively, the phenomenon
    is also explained in terms of bifurcation of stationary patterns as well
    as in the lack of a certain stability. All related mathematical proofs
    are based on degree theory.
  • 14.05.13   10:30

  • Eduard Feireisl
  • (Institute of Mathematics, CAS)
  • Scale interactions in compressible rotating fluids
  • Abstract:
  • We consider the motion of a viscous compressible rotating fluid confined to an infinite horizontal slab. We perform the singular limit for low Mach number (incompressible limit), low Rossby number (fast rotation limit) and high Reynolds number (inviscis limit). The limit problem will be identified as the 2D incompressible Euler system.
  • 30.04.13   09:00

  • Jens Frehse
  • (Institut fur Angewandte Mathematik, Universitat Bonn)
  • Selected question on elliptic regularity, On the multiplier problem for the Prandtl Reuss model.
  • 23.04.13   09:00

  • Jiri Neustupa
  • (Institute of Mathematics, CAS)
  • A refinement of some local regularity criteria for weak solutions to the Navier-Stokes equations
  • Abstract:
  • There exist a series of local regularity criteria for weak solutions to the Navier-Stokes equations, that state that the solutions is regular at the space-time point (x,t) if it satisfies some conditions in a backward space-time neighbourhood of (x,t) - typically, if the velocity has some rate of integrability in a backward parabolic neighbourhood of (x,t). We show that it is sufficient to impose conditions on the velocity only in the exterior of a certain space-time paraboloid with vertex at point (x,t), intersected with a backward neighbourhood of (x,t). We discuss further possible extensions of this criterion.
  • 16.04.13   09:00

  • Jan Burczak
  • (Faculty of Mathematics and Physics, Charles University)
  • Almost everywhere Holder continuity of gradients to non-diagonal parabolic systems
  • Abstract:
  • I present a local almost everywhere $C^{1,alpha}$-regularity
    result for a general class of p-nonlinear non-diagonal parabolic systems. The main part of the considered systems depends on space-time variable, solution and symmetric part of the gradient of solution. To obtain our result, I adapt for the symmetric-gradient case techniques developed for the full-gradient case by Mingione and coauthors.
  • 09.04.13   09:00

  • Miroslav Bulicek
  • (Faculty of Mathematics and Physics, Charles University)
  • Threshold slip and non-Newtonian incompressible fluids
  • Abstract:
  • In the modeling of the flow of a fluid the important role plays the prescribed boundary conditions for the velocity. We consider the internal flow and discuss the physically relevant boundary conditions for the velocity (no-slip, perfect slip, Navier's slip). Unfortunately, none of them is in perfect agreement with the reality and we propose "threshold slip" boundary conditions, which seems to be the most appropriate choice. We also discuss the key difference between no-slip and threshold slip from the point of view of analysis. In particular, we show the existence of an integrable pressure for the threshold slip which is not known for the no-slip boundary conditions.
  • 02.04.13   09:00

  • Ondrej Kreml
  • (Institute of Mathematics, CAS)
  • Global ill-posedness of the compressible isentropic Euler equations
  • Abstract:
  • We consider the isentropic compressible Euler system in 2 space
    dimensions with pressure law $p (rho) = rho^2$ and we show the existence
    of Lipschitz initial data for which there are infinitely many global
    bounded admissible weak solutions. The proof is based on recent results of
    De Lellis and Székelyhidi on incompressible Euler equations.
  • 26.03.13   09:00

  • Antonín Novotný
  • (IMATH, Universite du Sud Toulon-Var)
  • Dissipative solutions and singular limits in the thermodynamics of viscous fluids
  • Abstract:
  • We shall investigate the interaction of scales in the complete
    Navier-Stokes-Fourier system
    including low Mach number and high Reynolds and Peclet numbers with
    the general initial data.
    The limit system will be identified as the incompressible Euler system
    for the velocity field coupled
    with the transport equation for the temperature. It is a joint work
    with Eduard Feireisl.
  • 19.03.13   09:00

  • Francesco Fanelli
  • (BCAM, Spain)
  • Some models of non-homogeneous inviscid fluids
  • Abstract:
  • In this talk I will present some mathematical models for non-homogeneous prefect fluids. In a first time, we will focus on the incompressible case, i.e. on the density-dependent incompressible Euler system. We will show results on well-posedness in the class of Besov spaces and on propagation of geometric properties, such as striated and conormal regularity. These properties are a natural way of generalizing the classical 2-D vortex patches structure for homogeneous fluids. In the final part of the talk, we will deal with an inviscid low-Mach number limit system for compressible perfect fluids with heat conduction. Again, we will investigate problems like well-posedness in Besov spaces, continuation criteria, lifespan of the solution. Despite the difference of the model, the results are similar to those for the non-homogeneous incompressible Euler equations.
  • 12.03.13   10:15

  • Donatella Donatelli
  • (University of L'Aquila, Italy)
  • Analysis of Oscillations and Defect Measures in Plasma Physics
  • Abstract:
  • We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier Stokes Poisson system in 3 − D. We show that as the Debye length goes to zero the velocity field strongly converges towards an incompressible velocity vector field and the density fluctuation weakly converges to zero. In general the limit velocity field cannot be expected to satisfy the incompressible Navier Stokes equation, indeed the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self interacting wave packets. We shall provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis.
  •                    09:00

  • Lu Yong
  • (Universite Paris Diderot - Paris 7)
  • A stability criterion for high-frequency oscillations
  • Abstract:
  • We show that a simple compatibility condition determines the qualitative behavior of the solutions to semilinear, dispersive, hyperbolic initial-value problems issued from highly-oscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that interactions coefficients are not too large at the resonances.

    If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time.

    The amplification mechanism is based on the observation that in frequency space, resonances correspond to points of weak hyperbolicity. At such points, the behavior of the system depends on the lower order terms through the compatibility condition.

    Our examples include coupled Klein-Gordon equations in semilinear
    case.
  • 26.02.13   09:00

  • Pavol Quittner
  • (Comenius University, Bratislava)
  • Symmetry of components for semilinear elliptic systems
  • Abstract:
  • We give sufficient conditions ensuring that any positive classical solution (u,v) of an elliptic system in the whole n-dimensional space has the symmetry property u=v. We also provide several counterexamples which indicate that our assumptions are in a sense necessary. We are particularly interested in elliptic systems of Schrodinger type.
  • 15.01.13   09:00

  • Peter Takáč
  • (University of Rostock)
  • Space-time analyticity of $L^2$-solutions of parabolic systems (applications to finance)
  • 08.01.13   11:30

  • David Krejčiřík
  • (Nuclear Physics Institute AS CR)
  • The improved decay rate for the heat semigroup with local magnetic field in the plane
  • Abstract:
  • We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta.

    The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schroedinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behaviour of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov-Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables.
  •                    10:15

  • Peter Wittwer
  • (Université de Genčve)
  • A space-time approach to periodic solutions of the Navier-Stokes equations with drift
  • Abstract:
  • We construct solutions for the Navier-Stokes equations in three dimensions with a time periodic force which is of compact support in a frame that moves at constant speed. These solutions are related to solutions of the problem of a body which moves within an incompressible fluid at constant speed and rotates around an axis which is aligned with the motion. Our analysis is based on the use of function spaces which incorporate in a natural way the space-time structure of the solutions.
  •                    09:00

prof. RNDr. Eduard Feireisl, DrSc.
Šárka Nečasová, Milan Pokorný
chairmen