Asymptotic stability of composite waves of two viscous shocks for relaxed compressible Navier-Stokes equations

TL;DR

This paper proves the asymptotic stability of composite viscous shock waves in relaxed compressible Navier-Stokes equations, showing convergence to classical solutions as relaxation vanishes, using energy and entropy methods.

Contribution

It establishes the nonlinear stability of composite shock waves in a relaxed system and their convergence to classical solutions as the relaxation parameter tends to zero.

Findings

Composite waves with two shocks are asymptotically stable.

Solutions converge to classical system solutions as relaxation parameter approaches zero.

Stability proven under small, independent wave strengths and minor initial perturbations.

Abstract

This paper investigates the time asymptotic stability of composite waves formed by two shock waves within the context of one-dimensional relaxed compressible Navier-Stokes equations. We demonstrate that the composite waves consisting of two viscous shocks achieve asymptotic nonlinear stability under the condition of having two small, independent wave strengths and the presence of minor initial perturbations. Furthermore, the solutions of the relaxed system are observed to globally converge over time to those of the classical system as the relaxation parameter approaches zero. The methodologies are grounded in relative entropy, the $a$-contraction with shifts theory and fundamental energy estimates.