Seminar

  • 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

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