Seminar

  • Giselle Monteiro
  • (Institute of Mathematics, CAS)
  • On Preisach operators and piezoelectricity modeling
  • Abstract:
  • Preisach operators are rate independent hysteresis operators capable of reproducing minor loops, therefore they are well fitted to measurements of smart materials. Benefiting from this observation, some authors have proposed models for piezoelectricity assuming that all hysteresis effects are due to one single Preisach operator. More accurate models though have to account thermal effects. To address this problem, we introduce a notion of parameter-dependent Preisach operator and investigate some properties of its inverse.
    This is a joint work with P. Krejci.
  • 11.12.18   09:00

  • Martin Fencl
  • (University of West Bohemia, Pilsen)
  • Unilateral sources of an activator in reaction-diffusion systems describing Turing’s patterns
  • Abstract:
  • See the attachment.
  • 04.12.18   10:00

  • Ondřej Kreml
  • (Institute of Mathematics, CAS)
  • Wild solutions for isentropic Euler equations starting from smooth initial data
  • Abstract:
  • In a series of papers starting with the groundbreaking work of De Lellis and Székelyhidi several authors have shown that there might exist infinitely many bounded weak solutions to the isentropic Euler equations satisfying the energy inequality and starting from certain class of initial data. Concerning smoothness, the best result is due to Chiodaroli, De Lellis and Kreml, where the existence of these wild solutions was shown for Lipschitz initial data. In this talk we present the same result for smooth initial data. The proof is based on a nontrivial generalization of the previous theorem, in particular on a notion of generalized fan subsolution.
    This is a joint work with E. Chiodaroli, V. Mácha and S. Schwarzacher.
  • 27.11.18   09:00

  • Xavier Blanc
  • (Université Paris-Diderot)
  • Homogenization in the presence of defects
  • Abstract:
  • We will present some results on homogenization for linear elliptic equation. The geometry will be assumed to be either a perturbation of a periodic background, or an interface between two periodic media. In both cases, we study the homogenization problem, prove existence of a corrector, and use to build a two-scale expansion of the solution. We prove convergence estimates of this two-scale expansion.
    These are a joint works with C. Le Bris (Ecole des Ponts, Paris), P.-L. Lions (Collčge de France, Paris) and M. Josien (Ecole des Ponts, Paris).
  • 20.11.18   09:00

  • Bangwei She
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Convergence of a finite volume scheme for the compressible Navier-Stokes system
  • Abstract:
  • We study the convergence of a finite volume scheme for the compressible (barotropic) Navier--Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system.  Then by the weak-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.
    This is a joint work with E. Feireisl, M. Lukacova and  H. Mizerova.
  • 13.11.18   09:00

  • Václav Mácha
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Body with a Cavity Filled with a Compressible Fluid
  • Abstract:
  • We  study  the  dynamics  of  a  system  composed  by a rigid body containing a visous compressible fluid. The  emphasis  is  laid  upon  the  analysis  of  the  long time behavior of the whole system.  We show that for small initial data the whole system tends to a permanent rotation similarly as in the incompressible case. On the other hand, we highlight some problems coming from compressibility which do not allow to prove the same for solutions emanating from arbitrary initial data.
    The work was done in collaboration with G. P. Galdi and S. Necasova
  • 30.10.18   09:00

  • Ansgar Jüngel
  • (TU Wien)
  • Analysis of diffusive population systems for multiple species
  • Abstract:
  • The dynamics of multi-species populations can be described by random walks on a lattice which leads in the diffusive limit to nonlinear reaction-cross-diffusion systems. A special model was suggested by Shigesada, Kawasaki, and Teramoto in 1979. The diffusion matrix of these cross-diffusion systems is typically neither symmetric nor positive definite, which complicates the analysis. The idea is to reveal a so-called entropy structure (which is a special Lyapunov functional) allowing for gradient estimates. In this talk, we review recent results on population cross-diffusion models, including the local and global existence analysis, uniqueness of weak solutions, and their large-time asymptotics.
  • 23.10.18   09:00

  • Tong Tang
  • (Institute of Mathematics, Czech Academy of Sciences)
  • On a singular limit for the stratified compressible Euler system
  • Abstract:
  • We consider a singular limit for the compressible Euler system in the low Mach number regime driven by a large external force. We show that any dissipative measure-valued solution approaches a solution of the lake equation in the asymptotic regime of low Mach and Froude numbers. The result holds for the ill-prepared initial data creating rapidly oscillating acoustic waves. We use dispersive estimates of Strichartz type to eliminate the effect of the acoustic waves in the asymptotic limit.
  • 16.10.18   09:00

  • Tomasz Piasecki
  • (University of Warsaw)
  • Strong solutions to the steady compressible Navier-Stokes equations with inflow boundary conditions
  • Abstract:
  • We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition in a vicinity of given laminar solutions under the assumption that the pressure is a linear function of the density. In particular, we do not require any information on the gradient of the density or second gradient of the velocity. Our result holds provided certain condition on the shape of the boundary around the points where characteristics of the continuity equation are tangent to the boundary, which holds in particular for piecewise analytical boundaries.
  • 02.10.18   09:00

  • Jiří Neustupa
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Spectral instability of a steady flow of an incompressible viscous fluid past a rotating obstacle
  • Abstract:
  • We show that a steady solution U to the system of equations of motion of an incompressible Newtonian fluid past a rotating body is unstable if an associated linear operator L has at least one eigenvalue in the right half-plane in C. Our theorem does not directly follow from a series of preceding results on instability, mainly because the associated nonlinear operator is not bounded in the same space in which the instability is studied. As an important auxiliary result, we also show that the uniform growth bound of the C_0 semigroup e^{Lt} is equal to the spectral bound of operator L.
  • 15.05.18   09:00

  • Lucio Boccardo
  • (Sapienza University of Rome)
  • Regularizing effect of the lower order terms in some nonlinear Dirichlet problems
  • Abstract:
  • See the attachment.
  • 11.05.18   10:00

  • Zdeněk Skalák
  • (Czech Technical University)
  • Regularity criteria for the Navier-Stokes equations in terms of the velocity gradient
  • Abstract:
  • See the attachment.
  • 24.04.18   09:00

  • Petr Pelech
  • (Institute of Information Theory and Automation, Czech Academy of Sciences)
  • Gradient Polyconvexity in the Framework of Rate-Independent Processes
  • Abstract:
  • The talk treats mathematical aspects of evolutionary material models for shape-memory alloys at finite-strains. The difficulty of related mathematical analysis consists in the non-linear and non-convex dependence of the energy on the deformation gradient. One possible way, how maintain the analysis tractable, is to suppose that the energy depends also on the second deformation gradient and is convex in it. We relax this assumption by using the recently proposed concept of gradient-polyconvexity. Namely, we consider energies which are convex only in gradients of non-linear minors (i.e. cofactor and determinant in three dimension) of the deformation gradient. As a result, the whole second deformation gradient needs not to be integrable. Yet, at the same time, the obtained compactness is sufficient and, moreover, additional physically desirable properties(e.g. local invertibility) can be shown. We extend the previous result for hyperelastic materials by incorporating a rate-independent dissipation to our model and by proving existence of an energetic solution to it. It is a joint work with Martin Kružík (Institute of Information Theory and Automation of the Czech Academy of Sciences) and Anja Schloemerkemper (University of Wuerzburg).
  • 10.04.18   09:00

  • Miroslav Bulíček
  • (Charles University)
  • Nonlinear ellitpic and parabolic 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 radial structrure, which is a natural class of operators in the theory of PDE. As a by product we develop new theoretical tools as e.g., weighted estimates for the linear problems and the new compensated compactness method represented by the div-curl-biting-weighted lemma.
  • 27.03.18   09:00

  • Simon Markfelder
  • (Julius-Maximilians-Universität Würzburg)
  • Non-uniqueness of entropy solutions to the 2-d Riemann problem for the Euler equations
  • Abstract:
  • In this talk we consider the compressible (full) Euler equations in two space dimensions together with Riemann initial data. The issue of the talk is the question on uniqueness of weak entropy solutions to this problem. This issue has been studied for the isentropic Euler equations by E. Chiodaroli, C. De Lellis and O. Kreml (among others) and the aim is now to extend the results to full (i.e. non-isentropic) Euler. We consider a special class of Riemann data, namely those for which the 1-d self-similar solution consists of two shocks and possibly a contact discontinuity. We show that for this class there exist infinitely many weak entropy solutions, which are generated by convex integration. This is joint work with H. Al Baba, C. Klingenberg, O. Kreml and V. Mácha.
  • 20.03.18   09:00

  • Jan Březina
  • (Tokyo Institute of Technology)
  • Measure-valued solutions and Navier-Stokes-Fourier system
  • Abstract:
  • Encouraged by the ideas and results obtained when studying measure-valued solutions for the Complete Euler system we introduce measure-valued solutions to the Navier–Stokes–Fourier system and show weak-strong uniqueness. Namely, we identify a large class of objects that we call dissipative measure–valued (DMV) solutions, in which the strong solutions are stable. That is, a (DMV) solution coincides with the strong solution emanating from the same initial data as  long as the latter exists.
  • 13.03.18   09:00

  • Erika Maringová
  • (Charles University)
  • Up to the boundary Lipschitz regularity for variational problems
  • Abstract:
  • We prove the existence of a regular solution to a wide class of convex, variational integrals. Via technique of construction of the barriers we show that the  solution is Lipschitz up to the boundary. For the linear growth case, we identify the necessary and sufficient condition to existence of solution; in the case of superlinear growth, we provide the sufficient one. The result is not restricted to any geometrical assumption on the domain, only its regularity plays the role.
  • 06.03.18   09:00

  • Gabrielle Brüll
  • (Norwegian University of Science and Technology)
  • Weak solutions to a two-phase thin film equation with insoluble surfactant
  • Abstract:
  • We discuss a model describing the spreading of an insoluble surfactant on the upper surface of a viscous complete wetting two-phase thin film. Considering capillary effects as the only driving force, the system of evolution equations consists of two strongly coupled degenerated equations of fourth order describing the film heights of the fluids, which are additionally coupled to a second-order transport equation for the surfactant concentration. Owing to the degeneracy, it is in general not clear whether one can prove the existence of global solutions in a classical sense, which motivates the study of weak solutions. The proof of existence of nonnegative global weak solutions is based on a priori energy estimates and compactness arguments.
  • 27.02.18   09:00

  • Dalibor Pražák
  • (Charles University)
  • Regularity and uniqueness for a critical Ladyzhenskaya fluid
  • Abstract:
  • We consider an incompressible p-law type fluid in a 3D bounded domain. Employing iterative estimate in Nikolskii spaces and reverse Hölder inequality, we establish higher time regularity and uniqueness of weak solution provided the data are more regular.
    This is a joint work with M. Bulíček and P. Kaplický.
  • 20.02.18   09:00

  • Petr Kučera
  • (Czech Technical University)
  • Some properties of the strong solution of the Navier-Stokes equations
  • Abstract:
  • TBA
  • 09.01.18   09:00

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