Stability of a Riemann Shock in a Physical Class: From Brenner-Navier-Stokes-Fourier to Euler

The stability of an irreversible singularity, such as a Riemann shock solution to the full Euler system, in the absence of any technical conditions for perturbations, remains a major open problem even within a mono-dimensional framework. A natural approach to justify the stability of such a singularity involves considering a class of vanishing physical dissipation limits (or viscosity limits) of physical viscous flows with evanescent viscosities. We prove the existence of vanishing physical dissipation limits, on which a Riemann shock of small jump strength is stable (up to a time-dependent shift) and unique. As a physical viscous model that generates vanishing dissipation limits, we adopt the Brenner-Navier-Stokes-Fourier system, proposed by Brenner based on the bi-velocity theory. For this viscous system, we show the uniform stability of shock with respect to the strength of the viscosity. The uniformity is ensured by the contraction estimates of any large perturbations around the shock. Since we cannot impose any smallness on initial perturbations for the contraction estimates, the density could be arbitrarily small or large. Controlling such large perturbations is the most challenging part in our analysis. The contraction estimates ensure the existence of vanishing physical dissipation limits of solutions to the Brenner-Navier-Stokes-Fourier system. In addition, the distance of those limits from the Riemann shock, up to shifts, is controlled by the magnitude of the initial perturbations. The distance is measured by the relative entropy with a weight depending on the shock. This is the first result for the stability of Riemann shock solution to the full Euler system in a class of vanishing physical dissipation limits.