Seminar

Matteo Caggio
 (Institute of Mathematics, CAS)

NavierStokes equations and related problems in mathematics and physics

Abstract:

We will deal with NavierStokes equations for compressible and incompressible fluids and we will discuss some related problems in mathematics and physics. In the mathematical context, we will discuss singular limits problems for compressible fluids, fluids subjected to dimension reduction and conditional regularity for incompressible NavierStokes equations in the whole space. Some working progress in the context of RayleighBenard problem for micropolar fluids, weakstrong uniqueness for fluidstructure interaction and thinlayer approximation will be also presented. In the physical context, we will discuss problems related to the turbulent boundary layer. In particular, the CamassaHolm equations for a flatplate boundary layer and the similarity theory for a stationary atmospheric surfacelayer over horizontally homogeneous terrain.

Ond?ej Kreml
 (Institute of Mathematics, CAS)

Recent progress in the analysis of the Riemann problem for the 2D compressible Euler equations

Abstract:

The Riemann problem presents a basic tool in the analysis of hyperbolic conservation laws in 1D. Recently, it has been used also to demonstrate the ill behaviour of incompressible and compressible Euler systems in multiple dimensions, where the convex integration theory of De Lellis and Székelyhidi can be applied. Apart from being just a tool, the Riemann problem for compressible isentropic Euler system itself provides an interesting problem. It can be shown that for some Riemann data the problem admits a unique solutions whereas for different data nonuniqueness appears. Therefore, a natural question arises: Where is the threshold between uniqueness and nonuniqueness of admissible weak solutions? We will review the history of results yielding the answer with one open problem still remaining. Further we take a look on the full Euler system and present some conjectures based on data observations.
prof. RNDr. Eduard Feireisl, DrSc.
árka Nečasová, Milan Pokorný
chairmen