Seminar on partial differential equations

We discuss the limiting behaviour ('Gamma limit') of the 3d nonlinear elasticity theory for thin shells around 2d surfaces of arbitrary geometry. In particular, under the assumption that the elastic energy of deformations scales like (thickness of the shell)^4, we derive a limiting theory for 2 dimensional surfaces which is a generalization of the von Karman theory for plates.

In these lectures we consider the problem of maximal regularity for evolution equations in the context of $L^p$-spaces. We will discuss two approaches to this problem, one by singular integrals and the other one by operator sums. Topics include the vector-valued version of Mikhlin's theorem, the characterization of maximal regularity in terms of $R$-boundedness of the resolvent as well as the Dore-Venni theorem. The results will be applied to the Stokes system and to certain fluid-structure interaction problems.

Our purpose is to show existence of weak solutions to steady flow of non-Newtonian incompressible fluids with nonstandard growth conditions of the Cauchy stress tensor. We are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening. Since $L^p$ framework is not sufficient to capture the described model, we describe the growth conditions with help of general $x$--dependent convex function and formulate our problem in generalized Orlicz spaces. We will use the generalization of classical Minty-Browder method to nonreflexive spaces.

Finite-element approximation procedure will be presented for the system of equations arising from large eddy simulation of turbulent flows. We will show that solutions to the approximate problems generate strongly convergent subsequence whose limit is the weak solution of the original problem. To prove the convergence theorem we will use Young measures and related tools.

Na pracovním semináři vystoupí Jan Březina a Marek Kobera.

Part II - Leray-Alpha and Bardina models in the periodic case

We establish various estimates for norms with polynomial weights of weak and strong solutions to the Navier-Stokes equations in a 3D exterior domain. The results obtained improve the known results deduced so far by various authors and seem to be optimal in general.

Part IV - What remains true when one works with realistic BC such as the interaction between the ocean and the atmosphere, some open problems.

Na pracovním semináři vystoupí Martin Lanzendörfer.

Vít Průša: Stability of pipe flow
Libor Pavlíček: Reduced Sobolev spaces

An explicit formula for the fundamental solution of the linearized equations of fluid mechanics will be presented. These equations are obtained by linearizing the Navier-Stokes equations around a velocity that could be purely translational, resulting in the Stokes or Oseen equation, or include an angular velocity. The methods to obtain these solutions differ when these problems are considered in two or three dimensions. I will survey some of these aspects and present some properties of the resulting fundamental solutions. The talk is based on joint work with Ron Guenther.

We are interested by the contacts between rigid bodies moving into a viscous incompressible fluid. We consider the system composed by a rigid ball moving into a viscous incompressible fluid, over a fixed horizontal plane. The equations of motion for the fluid are the Navier--Stokes equations and the equations for the motion of the rigid ball are obtained by applying Newton's laws. We show that for any weak solutions of the corresponding system satisfying the energy inequality, the rigid ball never touches the plane.

Abstract: short long

Periodic homogenization means replacing an equation with periodic coefficients by an 'equivalent' constant coefficient equation which yields globally the same solution. The mathematic approach studies a sequence of problems with coefficients with diminishing period. The contribution aims to give basic ideas of a new approach to non-periodic homogenization developed by G. Nguetseng. It is based on the spectrum of Banach algebras and a new $\Sigma$-convergence.

Ondřej Kreml: Local-in-time existence for the FENE dumbbell model

The aim of these lectures is to present a reasonably self-contained theory of ergodicity for stochastic processes that is sufficiently flexible to allow to deal with infinite-dimensional problems like the stochastic Navier- Stokes equations, stochastic reaction-diffusion equations, etc. In the first lecture, we will introduce the main objects and problems, and remind the audience of the 'classical' theory of Harris chains. We will go through elementary sketches of proofs of some of the main results of this theory. In the second lecture, we will argue that the theory of Harris chains is not suitable for infinite-dimensional problems and we will lay down the foundations for a modified theory that is more flexible. The remainder of the course will be devoted to the applications of this theory to a class of stochastic PDEs. In the third lecture, we will sketch the proof of a general ergodicity result. The final lecture will be devoted to showing how to leverage the bounds obtained in the third lecture to obtain an exponential convergence result.