Seminar

  • 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

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