Colloque en l'honneur des 60 ans de Jérôme Pousin

Antonin Novotny

IMATH, Université du Sud Toulon-Var

Stability analysis for Compressible Navier-Stokes equations: theory and numerics.

In the first part of the talk, we introduce the notion of relative energy functional which measures a "distance" of a weak solution to the compressible Navier-Stokes equations to any other sufficiently smooth state of the fluid. We reformulate the thermodynamic stability conditions for the compressible fluids in terms of the relative energy inequality which describes the evolution of the relative energy functional. This process relies very much on the structure of the equations. The relative energy functional encodes most of the stability properties of the weak solutions to the compressible Navier-Stokes equations. In the second part of the talk, we adapt the above procedure to some FV/FE and FV/FD numerical schemes calculating compressible flows in order to investigate: 1) Uniform error estimates of the discrete numerical solution with respect to a strong solution. 2) The uniform stability of these numerical schemes in the low Mach number regime. The first part of the talk is based on several joint papers with E. Feireisl with contribution of B.J. Jin and Y. Sun. The second part of the talk is based on several joint works with T. Gallouet, R. Herbin, D. Maltese. continued by Feireisl, Hosek, Necasova, Lukacova, She.