Seminar

  • 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

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