Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system

TL;DR

This paper proves the time-asymptotic stability of superpositions of weak planar viscous shocks and rarefactions in multi-dimensional viscous systems, extending previous 1-D results to more complex systems.

Contribution

It extends the stability analysis of wave superpositions from 1-D to general multi-D $n\times n$ viscous systems using the $a$-contraction method.

Findings

Established stability of wave superpositions in multi-D systems.

Developed techniques for classifying and controlling higher-order terms.

Applied energy-based methods to complex wave interactions.

Abstract

We prove the time-asymptotic stability of the superposition of a weak planar viscous 1-shock and either a weak planar n-rarefaction or a weak planar viscous n-shock for a general $n\times n$ multi-D viscous system. In 2023, Kang-Vasseur-Wang [11] showed the stability of the superposition of a viscous shock and a rarefaction for 1-D compressible barotropic Navier-Stokes equations and solved a long-standing open problem officially introduced by Matsumura-Nishihara [23] in 1992. Our work is an extension of [11], where a general $n\times n$ multi-D viscous system is studied. Same as in [11], we apply the $a$-contraction method with shifts, an energy based method invented by Kang and Vasseur in [9], for both viscous shock and rarefaction at the level of the solution. In such a way, we can work with general perturbations and compositions of waves. Finally, a technique to classify and control higher-order terms is developed to work in multi-D.