Seminar

  • Camillo De Lellis
  • (Universität Zürich )
  • Approaching Plateau's problem with minimizing sequences of sets
  • Abstract:
  • In a joint paper with Francesco Ghiraldin and Francesco Maggi we provide a compactness
    principle which is applicable to different formulations of Plateau's problem in codimension one
    and which is exclusively based on the theory of Radon measures and elementary comparison
    arguments. Exploiting some additional techniques in geometric measure theory, we can use
    this principle to give a different proof of a theorem by Harrison and Pugh and to answer a
    question raised by Guy David.
  • 15.12.15   09:00

  • Milan Pokorny
  • (Charles University in Prague )
  • Heat-conducting, compressible mixtures with multicomponent diffusion
  • 08.12.15   09:00

  • Tomasz Piasecki
  • (Institute of Mathematics of the Polish Academy of Sciences)
  • Stationary compressible Navier-Stokes Equations with inflow boundary
  • Abstract:
  • The existence of solutions to the stationary compressible Navier-Stokes equations with boundary conditions admitting inflow and outflow is in general open question. All known results require some assumptions on smallness of the data and additional conditions on the shape of the boundary. I will discuss briefly known results of this type and show a new estimate in fractional order Sobolev spaces which seems new and promising approach in stationary case. This is a joint result with Piotr Mucha.
  • 01.12.15   09:00

  • Jiri Jarusek
  • (IM CAS)
  • Rational contact model with finite interpenetration
  • Abstract:
  • The classical Signorini contact model respects the impenetrability of Mass. However, the microscopic structure of every material even that with macroscopically seemingly perfect surface has small deformable asperities and/or holes to be filled. The model to be presented allows some surface interpenetration between the body and the contacted support. Unlike "normal compliance" approach sometimes used, the interpenetration is strictly limited here and its limit is not reachable outside a zero measure. The static version of the model for both frictionless and frictional contact will be treated together with the dynamic frictionless version.
  • 24.11.15   09:00

  • Jiří Neustupa
  • (IM CAS)
  • A spectral criterion for stability of a steady flow in an exterior domain
  • Abstract:
  • We show that the question of stability of a steady incompressible Navier-Stokes flow V in a 3D exterior domain depends on the time-decay of a finite family of concretely defined functions. Then, although the associated linearized operator has an essential spectrum touching the imaginary axis, we show that certain assumptions on the eigenvalues of this operator guarantee the stability of flow V, regardless the essential spectrum. No assumption on the smallness of the steady flow V is required.
  • 10.11.15   09:00

  • Filippo Dell'Oro
  • (IM CAS)
  • Energy inequality in differential form for weak solutions to the compressible Navier-Stokes equations on unbounded domains
  • Abstract:
  • We consider the Navier-Stokes equations of compressible isentropic viscous fluids on an unbounded three-dimensional domain with a compact Lipschitz boundary. Under the condition that the total mass of the fluid is finite, we show the existence of globally defined weak solutions satisfying the energy inequality in differential form. This is a joint work with E. Feireisl.
  • 03.11.15   09:00

  • Margarida Baía
  • (Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal)
  • A model for phase transitions with competing terms
  • 20.10.15   10:20

  • Elisabetta Chiodaroli
  • ( École Polytechnique Fédérale de Lausanne  )
  • A class of large global solutions for the Wave-Map equation
  • Abstract:
  • In this talk we consider the equation for equivariant wave maps from
    3+1- Minkowski space-time to the three dimensional sphere and we prove
    global in forward time existence of certain smooth solutions which have
    infinite critical Sobolev norm. Our method is based on a perturbative
    approach around suitably constructed approximate self-similar
    solutions.
  •                    09:00

  • Emil Wiedemann
  • (Hausdorff Center for Mathematics)
  • Weak-strong uniqueness in fluid dynamics
  • 13.10.15   10:20

  • Agnieszka Swierczewska-Gwiazda
  • (Faculty of Mathematics, Informatics and Mechanics, University of Warsaw)
  • On various compressible models of fluid mechanics: weak and measure-valued solutions
  • Abstract:
  • I will discuss the issue of existence of weak and measure-valued solutions to various systems of Euler type.  The most attention will be directed to the system of shallow water  type capturing flows of granular media, but I will also mention the pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity appearing in  collective behavior patterns.
  •                    09:00

  • Water Supply Interruption
  • Seminar is cancelled
  • 06.10.15   09:00

  • Filip Rindler
  • (University of Warwick)
  • Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms
  • Abstract:
  • !!! EXCEPTIONALLY AT SEMINAR ROOM K6, FACULTY OF MATHEMATICS AND PHYSICS, CHARLES UNIVERSITY IN PRAGUE, SOKOLOVSKA 83, PRAHA 8 !!!

    Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^p-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending both the theory of (generalized) Young measures and the theory of H-measures. Since in L^p-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, MCFs allow to quantify the difference between these two notions of compactness. The definition involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily.
    Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. Moreover, the hierarchy of oscillations with regard to slow and fast scales can be investigated as well since this information is also is reflected in the generated MCF.
  • 02.06.15   09:00

  • Vaclav Macha
  • Dynamics of a body containing a vicsous compressible fluid
  • 19.05.15   11:15

  • Jiri Neustupa
  • Stability of a steady flow of a viscous incompressible fluid past a fixed or rotating body
  •                    10:40

  • Patrick Penel
  • Mathematical gardening: two tools recently used in the theory of Navier-Stokes equations
  •                    09:40

  • Miloslav Feistauer
  • Analysis of discontinuous Galerkin method for PDEs with corner singularities
  •                    09:05

  • Petr Kaplicky
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On $L^q$ estimates of planar flows up to boundary
  • Abstract:
  • We show apriori $L^q$ gradient estimates for a sufficiently smooth planar flow driven by generalized Stokes system of equations. The estimates are obtained up to the boundary of a container where the fluid is contained. We allow power growth $p-1$, $pin(1,+infty)$ of the extra stress tensor for large values of shear rate but we exclude degeneracy or singularity for small shear rate. We also allow arbitrary $qin[p,+infty)$. The technique is based on a new type of Sobolev embedding theorem.
  • 12.05.15   09:00

  • Martin Väth
  • (Free University Berlin)
  • Stability in L_2, W^{1,2}: Extrapolation Spaces and Gel'fand Triples
  • Abstract:
  • Starting from the "naive" question how to prove stability of semilinear parabolic systems in the spaces L_2 and W^{1,2} by means of linearization, one is led to two different "natural" approaches. The aim of the talk is to compare both approaches and to reveal some connections, e.g. with Kato's square root problem.
  • 05.05.15   09:00

  • Simon Axmann
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On a result of D. Gerard-Varet and N. Masmoudi in homogenization
  • Abstract:
  • We will refer on the paper of David Gerard-Varet and Nader Masmoudi: Homogenization and boundary layers
  • 21.04.15   10:20

  • Jan Burczak
  • (Institute of Mathematics, Polish Academy of Sciences)
  • The Keller-Segel system
  • Abstract:
  • I will present one of the most popular PDEs stemming from the mathematical biology, namely the Keller-Segel system. The first part of my talk is devoted to presenting applicational motivation and classical rudiments of the analysis of the Keller-Segel system. In the second part, I will focus on Keller-Segel systems with general diffusions, including semilinear and fractional ones. In particular, a disproof of a blowup conjecture for the critical, fractional one-dimensional case will be sketched. This last part is based on a joint work with Rafael Granero (Davies).
  •                    09:00

  • Simon Axmann
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On a result of D. Gerard-Varet and N. Masmoudi in homogenization
  • Abstract:
  • We will start referring on the paper of David Gerard-Varet and Nader Masmoudi: Homogenization and boundary layers
  • 14.04.15   10:20

  • Radim Hosek
  • (Institute of Mathematics, Academy of Sciences of the Czech Republic)
  • Strongly regular family of meshes approximating C^2 domains for numerical method for compressible Navier-Stokes
  • Abstract:
  • In order to establish error estimates for a numerical method to compressible Navier-Stokes equations, we need a family of tetrahedral meshes covering approximative polyhedral domains $Omega_h$ satisfying $dist[partial Omega_h, partial Omega] < c h^2$, where $Omega$ is at least $C^2$ smooth bounded domain in 3 dimensional space.
    We show that having an initial mesh with certain properties, we can construct a strongly regular family of meshes.
  •                    09:00

  • Simon Axmann
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • On a result of D. Gerard-Varet and N. Masmoudi in homogenization
  • Abstract:
  • We will start referring on the paper of David Gerard-Varet and Nader Masmoudi: Homogenization and boundary layers.
  • 07.04.15   10:50

  • Manoj K. Yadav
  • (Faculty of Civil Engeneering, Czech Technical University in Prague)
  • On a result of N. Masmoudi in homogenization
  • Abstract:
  • We will finish referring on the paper of Nader Masmoudi: Homogenization of the compressible Navier-Stokes equations in a porous medium.
  •                    10:20

  • Sarka Necasova
  • (Institute of Mathematics, Academy of Sciences of the Czech Republic)
  • Singular limits in a model of compressible flow
  • Abstract:
  • We consider relativistic and "semi-relativistic" models of radiative viscous compressible Navier-Stokes-(Fourier) system coupled to the radiative transfer equation extending the classical model introduced in [1] and we study some of its singular limits in the case of well-prepared initial data and Dirichlet boundary condition for the velocity field see [2], [3], [4].

    References
    [1] B. Ducomet, E. Feireisl, S. Necasova: On a model of radiation hydrodynamics. Ann. I. H. Poincare - AN 28 (2011) 797–812.
    [2] B. Ducomet, S. Necasova: Diffusion limits in a model of radiative flow, to appear in Annali dell Universita di Ferrara, DOI 10.1007/s11565-014-0214-3.
    [3] B. Ducomet, S. Necasova: Singular limits in a model of radiative flow, to appear in J. of Math. Fluid Mech.
    [4] B. Ducomet, S. Necasova: Non equilibrium diffusion limit in a barotropic radiative flow, submitted to Volume Contemporary Mathematics Series of the American Mathematical Society; editors: Vicentiu Radulescu, Adelia Sequeira, Vsevolod A. Solonnikov.
  •                    09:00

  • Tomasz Piasecki
  • (Institute of Mathematics, Polish Academy of Sciences)
  • Stationary compressible Navier-Stokes equations with inflow boundary conditions
  • Abstract:
  • I will discuss recent results in the theory of compressible flows described by Navier-Stokes equations focusing on problems with inhomogeneous boundary data admitting inflow and outflow through the boundary. Admission of inflow /outflow leads to substantial mathematical difficulties. In particular there are no general existence results in the stationary case in such situation what motivates investigation of small data problems and simplified models.
  • 31.03.15   10:20

  • Stanislav Hencl
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • Jacobians of Sobolev homeomorphisms
  • Abstract:
  • In models of nonlinear elasticity people always assume that the deformation is orientation preserving, i.e. the Jacobian does not change sign. We show that for homeomorphisms in dimension n=3 we can assume this without loss of generality, i.e. each homeomorphism satisfies either J_fgeq 0 a.e. or J_fleq 0 a.e. This is a joint work with J. Maly.
  •                    09:00

  • Manoj K. Yadav
  • (Faculty of Civil Engeneering, Czech Technical University in Prague)
  • On a result of N. Masmoudi in homogenization
  • Abstract:
  • We will continue referring on the paper of Nader Masmoudi: Homogenization of the compressible Navier-Stokes equations in a porous medium.
  • 24.03.15   10:20

  • Ondrej Kreml
  • (Institute of Mathematics, Academy of Sciences of the Czech Republic)
  • Uniqueness of rarefaction waves in compressible Euler systems
  • Abstract:
  • We consider two systems of partial differential equations; the compressible isentropic Euler system and the complete Euler system describing the time evolution of an inviscid nonisothermal gas. In both cases we show that the rarefaction wave solutions of the 1D Riemann problem are unique in the class of all bounded weak solutions to the associated multi-D problem. This may be seen as a counterpart of the non-uniqueness results of physically admissible solutions emanating from 1D shock waves constructed recently by the method of convex integration.
  •                    09:00

  • Manoj K. Yadav
  • (Faculty of Civil Engeneering, Czech Technical University in Prague)
  • On a result of N. Masmoudi in homogenization
  • Abstract:
  • We will refer the paper of N. Masmoudi - Homogenization of the compressible Navier-Stokes equations in a porous medium.
  • 17.03.15   10:20

  • Miroslav Bulicek
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • Nonlinear elliptic equations beyond the natural duality pairing
  • Abstract:
  • Many real-world problems are described by nonlinear partial differential equations. A promiment example of such equations is nonlinear (quasilinear) elliptic system with given right hand side in divergence form div f data. In case data are good enough (i.e., belong to L^2), one can solve such a problem by using the monotone operator therory, however in case data are worse no existence theory was available except the case when the operator is linear, e.g. the Laplace operator. For this particular case one can however establish the existence of a solution whose gradient belongs to L^q whenever f belongs to L^q as well. From this point of view it would be nice to have such a theory also for general operators. However, it cannot be the case as indicated by many counterexamples. Nevertheless, we show that such a theory can be built for operators having asymptotically the Uhlenbeck structrure, which is a natural class of operators in the theory of PDE.
  •                    09:00

  • Bernard Ducomet
  • (CEA, DAM, DIF)
  • The P1 approximation for viscous barotropic and radiating flows
  • Abstract:
  • We consider a simplified model of radiative hydrodynamics consisting in coupling the barotropic Navier-Stokes system to the "P1" approximation of the radiative transfer equation. In the critical regularity setting, we prove global existence for data near a stable equilibrium. We also discuss some pertinent asymptotics (low Mach and diffusion limits). These results have been obtained in collaboration with Raphael Danchin (LAMA, Universite Paris Est).
  • 10.03.15   09:00

  • Martin Kalousek
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • Introductory lecture to homogenization problems
  • 03.03.15   10:20

  • Petr Kaplicky
  • (Faculty of Mathematics and Physics, Charles University in Prague)
  • BMO estimates for some elliptic problems
  • Abstract:
  • We will discuss BMO estimates for a weak solutions of the inhomogeneous p-Laplace system given by $-div(|nabla u|^{p-2} nabla u) = div f$. We show that $f in BMO$ implies $|nabla u|^{p-2} nabla u in BMO$, which is the limiting case of the nonlinear Calderon-Zygmund theory. This extends the work of DiBenedetto and Manfredi (1993), which was restricted to the super-quadratic case $pgeq 2$, to the full case $p in (1,infty)$ and even more general growth. We also briefly mention an application of the method to steady planar flows of generalized Newtonian fluids.
  •                    09:00

  • Marius Tucsnak
  • (Université de Lorraine)
  • Estimatability and observers for a model of population dynamics with diffusion and age dependence
  • Abstract:
  • We consider a model of McKendrick type for population dynamics
    with age dependence and diffusion. We prove that, Using various
    observation operators, we obtain an infinite system which is
    estimatable (detectable). This information is used to construct
    observers able to reconstruct population dynamics in the whole spatial
    domain from measures in an arbitrarily small spatial and age domain.
  • 27.01.15   09:00

  • Peter Takac
  • (Universitat Rostock)
  • Space-Time Analyticity of Solutions to Linear and Semilinear Parabolic Equations in the Whole Space with Applications to Two Volatility Models in Mathematical Finance
  • Abstract:
  • We begin by a brief presentation of a well-known mathematical model for the European option pricing in a market with stochastic volatility (J.-P. Fouque, G. Papanicolaou, and K. R. Sircar). We will use only intuitive probabilistic arguments in order to explain the main ideas of the arbitrage-free option pricing introduced in the works of F. Black and M. Scholes and (independently) R. C. Merton in 1973. These probabilistic arguments, combined with Ito's formula, yield an interesting parabolic partial differential equation (or a system of weakly coupled equations) for the option price(s). We briefly explain why, from the point of view of Mathematical Finance (complete markets), it is of interest to study a parabolic system of precisely N coupled scalar parabolic equations on an N-dimensional Euclidean space (or a cone).
    Then we proceed to the more technical, mathematical part of our lectures. We treat first the classical Black-Scholes-type model, i.e., a linear parabolic system for which we prove a theorem on the analytic smoothing property of a uniformly parabolic system with analytic coefficients, with only L^2-Lebesgue integrable initial data over R^N. The main difficulty in our approach will be to establish suitable a priori L^2-type estimates for the holomorphic extension of the weak solution to a complex strip around R^N. Such estimates are allowed to depend solely on the L^2-norm of the initial data over R^N. We will (have to) investigate how the width of the complex strip around R^N increases with (the real part of) time. We use the Hardy spaces of holomorphic functions to do this. The analyticity in time is obtained in a much more familiar way that takes advantage of the theory of holomorphic semigroups. This part is based on the author's recent work (2012).
    In the second lecture we treat a semilinear generalization of the linear model with fairly general analytic nonlinearities. We obtain the necessary uniform bounds (local in time and global in space) on the weak solution to the nonlinear system by an abstract (real) interpolation method combined with maximal regularity. To prove the time-analytic smoothing property, we adapt several ideas of S. Angenent (1990) to our functional L^p-setting in time. However, the global uniform bounds in space are obtained only for those initial data that themselves obey these bounds. This process takes place in an interpolation trace space of functions with fractional smoothness, i.e., in a Besov space. As this approach yields also a priori bounds for the weak solution, local in time and global in space, it is then easy to perform a linearization procedure to obtain a linear equation for the difference of two weak solutions with different initial data. The L^2-norm of this difference is thus controlled by the L^2-norm of the difference of the initial data on R^N. This result provides the same type of a priori estimate as for the linear system and is obtained in an analogous way. This part is based on the author's ongoing work with a Ph.D. student (2015).
  • 13.01.15   10:20

  • Emil Wiedemann
  • (Universitat Bonn)
  • Measure-Valued Solutions of the Euler Equations
  • Abstract:
  • Measure-valued solutions of the incompressible Euler equations were first considered by DiPerna and Majda to describe effects of oscillation and concentration in ideal fluids. Although measure-valued solutions appear a priori as much weaker objects than distributional solutions, we have been able to show that both notions are in a sense equivalent. An important open question concerns the relation between weak and measure-valued solutions for compressible Euler models. Joint work with L. Székelyhidi, Jr.
  •                    09:00

  • Peter Takac
  • (Institut fur Mathematik, Universitat Rostock)
  • Space-Time Analyticity of Solutions to Linear and Semilinear Parabolic Equations in the Whole Space with Applications to Two Volatility Models in Mathematical Finance
  • Abstract:
  • We begin by a brief presentation of a well-known mathematical model for the European option pricing in a market with stochastic volatility (J.-P. Fouque, G. Papanicolaou, and K. R. Sircar). We will use only intuitive probabilistic arguments in order to explain the main ideas of the arbitrage-free option pricing introduced in the works of F. Black and M. Scholes and (independently) R. C. Merton in 1973. These probabilistic arguments, combined with Ito's formula, yield an interesting parabolic partial di erential equation (or a system of weakly coupled equations) for the option price(s). We brie y explain why, from the point of view of Mathematical Finance (complete markets), it is of interest to study a parabolic system of precisely N coupled scalar parabolic equations on an N-dimensional Euclidean space (or a cone).
    Then we proceed to the more technical, mathematical part of our lectures. We treat first the classical Black-Scholes-type model, i.e., a linear parabolic system for which we prove a theorem on the analytic smoothing property of a uniformly parabolic system with analytic coefficients, with only L^2-Lebesgue integrable initial data over R^N. The main difficulty in our approach will be to establish suitable a priori L^2-type estimates for the holomorphic extension of the weak solution to a complex strip around R^N. Such estimates are allowed to depend solely on the L^2-norm of the initial data over R^N. We will (have to) investigate how the width of the complex strip around R^N increases with (the real part of) time. We use the Hardy spaces of holomorphic functions to do this. The analyticity in time is obtained in a much more familiar way that takes advantage of the theory of holomorphic semigroups. This part is based on the author's recent work (2012).
    In the second lecture we treat a semilinear generalization of the linear model with fairly general analytic nonlinearities. We obtain the necessary uniform bounds (local in time and global in space) on the weak solution to the nonlinear system by an abstract (real) interpolation method combined with maximal regularity. To prove the time-analytic smoothing property, we adapt several ideas of S. Angenent (1990) to our functional L^p-setting in time. However, the global uniform bounds in space are obtained only for those initial data that themselves obey these bounds. This process takes place in an interpolation trace space of functions with fractional smoothness, i.e., in a Besov space. As this approach yields also a priori bounds for the weak solution, local in time and global in space, it is then easy to perform a linearization procedure to obtain a linear equation for the difference of two weak solutions with different initial data. The L^2-norm of this difference is thus controlled by the L^2-norm of the difference of the initial data on R^N. This result provides the same type of a priori estimate as for the linear system and is obtained in an analogous way. This part is based on the author's ongoing work with a Ph.D. student (2015).
  • 06.01.15   09:00

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