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

Convergence of a finite volume scheme for the compressible Navier-Stokes system

Bangwei She
Institute of Mathematics, Czech Academy of Sciences
Tuesday, 13. November 2018 - 9:00 to 10:00
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.

Body with a Cavity Filled with a Compressible Fluid

Václav Mácha
Institute of Mathematics, Czech Academy of Sciences
Tuesday, 30. October 2018 - 9:00 to 10:00
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

Analysis of diffusive population systems for multiple species

Ansgar Jüngel
TU Wien
Tuesday, 23. October 2018 - 9:00 to 10:00
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.

On a singular limit for the stratified compressible Euler system

Tong Tang
Institute of Mathematics, Czech Academy of Sciences
Tuesday, 16. October 2018 - 9:00 to 10:00
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.