Seminar

  • Timofey Shilkin
  • (Max Planck Institute)
  • On elliptic equations with a singular drift from Morrey spaces
  • Abstract:
  • We investigate weak solutions to the Dirichlet problem for an elliptic equation with a drift b  whose divergence is sign-defined. We assume b belongs to some weak Morrey class which includes in the 3D case, in particular, drifts having a singularity along the axis with the asymptotic  c/r, where r  is the distance to the axis. The problem under consideration is motivated by some questions arising in the theory of axially symmetric solutions to the Navier-Stokes equations. We present results on existence, uniqueness and local properties of weak solutions to this problem as well as its relation to the Navier-Stokes theory. Based on a joint work with M. Chernobai.
  • 23.05.23   10:15

  • Yoshihiro Shibata
  • (Waseda University)
  • Local and global well-posedness of free boundary problem for the Navier-Stokes equations in exterior domains
  • Abstract:
  • See the attached lecture notes.
  •                    09:00

  • Yoshihiro Shibata
  • (Waseda University)
  • Maximal L_p-L_q theory for the Stokes equations with free boundary conditions
  • Abstract:
  • See the attached lecture notes.
  • 16.05.23   08:30

  • Yoshihiro Shibata
  • (Waseda University)
  • Introduction to free boundary problem for the Navier-Stokes equations and R-solver approach to this problem
  • Abstract:
  • See the attached lecture notes.
  • 09.05.23   09:00

  • Florian Oschmann
  • (Institute of Mathematics, CAS)
  • Some insights in homogenization of compressible Navier-Stokes equations
  • Abstract:
  • We investigate the homogenization of the two- and three-dimensional unsteady compressible Navier-Stokes equations, and show (i) the convergence result for the 2D case, and (ii) a better lower bound for the adiabatic exponent occurring in the pressure law for the 3D case. The latter will be covered with two different proofs. Lastly, we briefly discuss the optimality of the achieved bounds for the adiabatic exponent in terms of the underlying space-time dimension. This is joint work with Šárka Nečasová (CAS) and Milan Pokorný (Charles University).
  • 02.05.23   09:00

  • Maja Szlenk
  • (University of Warsaw)
  • A multifluid model with chemically reacting components - construction of weak solutions
  • Abstract:
  • We investigate the existence of weak solutions to the multi-component system, consisting of compressible chemically reacting components, coupled with the Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reaction and assume the nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum.

    It is shown that there exists a global weak solution, satisfying the $L^infty$ bounds for all the components. Moreover, despite the lack of the regularity on the gradients, we obtain strong compactness of densities in $L^2$. The applied method captures the properties of the models of high generality, which admit an arbitrary number of components. Furthermore, the framework we develop can handle models that contain both diffusing and non-diffusing elements.
  • 25.04.23   09:00

  • Petr Kaplický
  • (Charles University)
  • Stokes problems with dynamic boundary conditions
  • Abstract:
  • We show maximal regularity in time of solutions to the evolutionary Stokes problem with dynamic boundary condition in the case that the underlying space is Hilbert space.
  • 18.04.23   09:00

  • Eduard Feireisl
  • (Institute of Mathematics, CAS)
  • Compressible MHD as a dissipative system
  • Abstract:
  • We show that the compressible MHD system admits a bounded absorbing set in the energy ``norm'' as long as the open boundary conditions are imposed. In addition, the trajectories are precompact in a suitable topology. If this is the case, there is a compact global attractor as well as statistical stationary solutions supported by individual trajectories of weak solutions.
  • 11.04.23   09:00

  • Anna Abbatiello
  • (University of L'Aquila)
  • On the stability of incompressible heat conducting non-newtonian fluids
  • Abstract:
  • See the attached file.
  • 04.04.23   11:30

  • Nilasis Chaudhuri
  • (Imperial College)
  • Construction of the weak solutions to the barotropic Navier-Stokes system compatible with Kolmogorov compactness criterion
  • Abstract:
  • We prove the existence of the weak solutions to the compressible Navier--Stokes system with barotropic pressure $p(varrho)=varrho^gamma$ for $gammageq 9/5$ in three space dimensions. In the approximation scheme we use more direct truncation and regularisation of nonlinear terms and the pressure instead of the classical regularization of the continuity equation (based on the viscosity approximation $ep Delta varrho$). This scheme is compatible with the Kolmogorov (Bresch-Jabin) compactness criterion for the density. We revisit this criterion and prove that it can be applied in our approximation at any level.
  •                    10:15

  • Eva Fašangová
  • (TU Dresden)
  • The fractional Laplace operator (an introduction)
  • Abstract:
  • We motivate the heat equation with a fully discrete model (lattice gas cellular automaton), the classical as well as the one with the  fractional Laplace operator. We discuss the mathematical setting for the fractional case: the space (fractional Sobolev), the operator, the energy, boundary conditions, weak solution.
  •                    09:00

  • Thomas Eiter
  • (WIAS Berlin)
  • The concept of energy-variational solutions for hyperbolic conservation laws
  • Abstract:
  • We consider the notion of energy-variational solutions for hyperbolic conservation laws. This novel solvability concept is obtained by enriching the variational formulation by the weighted difference between the mechanical energy and an auxiliary variable representing the turbulent energy. If the weight is chosen suitably, an existence result for a general class of conservation laws can be derived via a time-discretization scheme based on a sequential minimization and, in particular, without a spatial regularization. The solution concept comes along with favorable properties like a weak-strong uniqueness principle and the convexity of solution sets. Moreover, for the compressible and incompressible Euler equations, energy-variational solutions can be identified with dissipative weak solutions.
  • 28.03.23   09:00

  • Antonín Češík
  • (Charles University)
  • Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision
  • Abstract:
  • The talk discusses existence theory for collisions of (visco-)elastic bulk solids which are undergoing inertial evolution. In particular, our approach for contact is based only on the assumption of non-interpenetration of matter. Most other theories for contact of elastic solids include some phenomenological assumptions, which we do not need in our approach.

    We are able to show existence of weak solutions including contact with an obstacle or with the solid itself, for arbitrarily large times and large deformations. Furthermore, our construction includes a characterization of the contact force which obeys conservation of momentum and an energy balance. This contact force is a vector-valued surface measure acting in the normal direction, and is constructed as a consequence of the non-interpenetration of matter.

    This is a joint work with Giovanni Gravina and Malte Kampschulte.
  • 21.03.23   09:00

  • Volodymyr Mikhailets
  • (Institute of Mathematics, CAS)
  • One-dimensional differential operators with distributions in coefficients
  • Abstract:
  • Some classes of linear ordinary differential operators with strongly singular coefficients are studied in the talk. These operators are introduced as quasi-differential according to Shin-Zettl. Their domains may not contain non-zero smooth functions. The case of self-adjoint Schrödinger and Hill operators on the line is investigated in more detail.
  • 14.03.23   09:00

  • Aneta Wróblewska-Kamińska
  • (Institute of Mathematics, Polish Academy of Sciences)
  • Relaxation limit of hydrodynamic models
  • Abstract:
  • We will show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We plan to discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We will deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we will mention how to show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.
    This is a joint result with Jose Carrillo and Yingping Peng.
  • 07.03.23   09:00

  • Ivan Gudoshnikov
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Elastoplasticity with softening in spring network models: a state-dependent sweeping process approach
  • Abstract:
  • Softening plasticity and Gurson model of damage in particular lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete failure of a material. Moreover, spatially continuous models with softening suffer from localization of strains and stresses to measure-zero submanifolds. We formulate a problem of quasistatic evolution of elasto-plastic spring networks (Lattice Spring Models) with a plastic flow rule which describes linear hardening, linear softening and perfectly plastic springs in a uniform manner. The fundamental kinematic and static characteristics of the network are described by the rigidity theory and structural mechanics.
    To solve the evolution problem we convert it to a type of a differential quasi-variational inequality known as the state-dependent sweeping process. We prove the existence of solution to the associated time-stepping problem (implicit catch-up algorithm), and the estimates we obtain imply the existence of a solution to the (time-continuous) sweeping process. Using numerical simulations of regular grid-shaped networks with softening we demonstrate the development of non-symmetric shear bands. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions, appearing in a bifurcation which happens when the parameters of the networks continuously change from hardening through perfect plasticity to softening.
  • 28.02.23   09:00

  • Sourav Mitra
  • (Institute of Mathematics, Czech Academy of Sciences)
  • Existence of weak solutions for a compressible multi-component fluid structure interaction problem
  • Abstract:
  • See the attached file.
  • 21.02.23   09:00

  • Matthieu Cadiot
  • (McGill University, Canada)
  • Rigorous Computation of Solutions of Semi-Linear Partial Differential Equations on Unbounded Domains Via Spectral Methods
  • Abstract:
  • In recent years, rigorous numerics have become a major tool to prove solutions of Partial Differential Equations (PDEs). However, when the equation is set on an unbounded domain of $mathbb{R}^m$ (where $m ge 1$), only a few results have been obtained so far. In this talk, I will present a general method to rigorously prove strong solutions to a large class of nonlinear PDEs in a Hilbert space $H^l subset H^s(mathbb{R}^m)$ ($sgeq 1$) via computer-assisted proofs.  We first introduce a method to rigorously compute an upper bound for the norm of the inverse of the linearization of PDE operators. The method is purely spectral and the constants are determined through Fourier analysis. Then using a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions to PDEs. I will illustrate the method using the special case of the Kawahara equation (fourth order KdV equation) and, as an application, I will present some computer-assisted proofs of localized patterns for the 2D Swift-Hohenberg equation.
  • 14.02.23   09:00

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