slideshow 3

Seminar on Partial Differential Equations

usually takes place each Tuesday at 09:00 in IM, rear building, ground floor
Chair: Šárka Nečasová, Milan Pokorný

Lipschitz bounds and non-uniform ellipticity

Lisa Beck
University of Augsburg
Tuesday, 26. November 2019 - 9:00 to 10:00
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).

Injective nonlinear elasticity via penalty terms: analysis and numerics

Stefan Krömer
Institute of Information Theory and Automation, CAS
Tuesday, 19. November 2019 - 9:00 to 10:00
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).

On the asymptotic limit of a shrinking source and sink in a 2D bounded domain

Marco Bravin
University of Bordeaux
Tuesday, 22. October 2019 - 9:00 to 10:00
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.

On a body with a cavity filled with compressible fluid

Václav Mácha
Institute of Mathematics, CAS
Tuesday, 15. October 2019 - 9:00 to 10:00
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.