Séminaire de EDP - Physique Mathématique

We derive an entropy stable extension of the Navier-Stokes-Fourier equations into the transition regime of rarefied gases. We do this through a variational multiscale reformulation of the closure of conservation equations derived from the Boltzmann equation. Our reformulation subsumes existing methods such as the Chapman-Enskog expansion. We apply the linearized version of this extension to the stationary heat problem and the Poiseuille channel and compare our analytical solutions to asymptotic and numerical solutions of the linearized Boltzmann equation. In both model problems, our solutions compare remarkably well in the transition regime. For some macroscopic variables, this agreement even extends far beyond the transition regime.

Pas de séminaire

We consider a 2 by 2 matrix-valued operator with two 1D Schroedinger operators on the diagonals and small in h interactions on the off-diagonals. We compute the semiclassical connection formula of the microlocal solutions (microlocal scattering matrix) at a crossing point of the two characteristic sets, and apply it to the resonance asymptotics. We will see that the subprincipal term of the microlocal scattering matrix is reduced toan osillatory integral whose critical point is the crossing point.