Uniform asymptotic stability for convection-reaction-diffusion equations in the inviscid limit towards Riemann shocks

TL;DR

This paper proves the uniform asymptotic orbital stability of viscous regularizations of Riemann shocks in scalar balance laws, with results valid across all viscosity levels and addressing multiscale spatial and spectral challenges.

Contribution

It introduces a novel uniform stability analysis for viscous shocks that does not rely on parabolic regularization, handling multiscale and spectral complexities.

Findings

Established uniform orbital stability of viscous shocks across viscosity parameters

Developed a multiscale analysis accommodating slow and fast spatial components

Designed a phase shift to encode stability without spectral gap assumptions

Abstract

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter $\epsilon$. The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale $\epsilon$-dependent topology reduces to the classical $W^{1,\infty}$-topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine-Hugoniot conditions and the non-uniform shift arising merely from phasing out the non-decaying $0$-mode, as in the classical stability analysis for fronts of reaction-diffusion equations.