Seminar

  • 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

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