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

TL;DR

This paper proves the stability and uniqueness of Riemann shocks in the full Euler system by analyzing vanishing dissipation limits of the Brenner-Navier-Stokes-Fourier model, overcoming previous open problems in shock stability.

Contribution

It establishes the first unconditional stability and uniqueness results for Riemann shocks in the full Euler system through a novel analysis of viscous shock limits.

Findings

Proved existence of vanishing dissipation limits for small amplitude shocks.

Showed Riemann weak shocks are stable and unique under physical disturbances.

Developed a robust method applicable to various models.

Abstract

The stability of an irreversible singularity, such as a Riemann shock to the full Euler system, in the absence of any technical conditions on perturbations, remains a major open problem even within mono-dimensional framework. A natural approach to justify such stability is to consider vanishing dissipation (or viscosity) limits of physical viscous flows. We prove the existence of vanishing dissipation limits, on which a Riemann shock of small amplitude is stable (up to a time-dependent shift) and unique. Thus, a Riemann weak shock is rigid (not turbulent) under physical disturbances. We adopt the Brenner-Navier-Stokes-Fourier system, based on the bi-velocity theory, as a physical viscous model. The key ingredient of the proof is the uniform stability of the viscous shock with respect to the viscosity strength. The uniformity is ensured by contraction estimates of any large perturbations around the shock. The absence of any restrictions on size of initial perturbations forces us to handle extreme values of density and temperature, which constitutes the most challenging part of our analysis. We use the method of a-contraction with shifts, but we improve it by introducing a more delicate analysis of the localizing effect given by viscous shock derivatives. This improvement possesses a degree of robustness that renders it applicable to a wide range of models. This is the first resolution for the challenging open problem on the "unconditional" stability and uniqueness of Riemann shock solutions to the full Euler system in a class of vanishing physical dissipation limits.