Seminar

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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