Asymptotic stability of a composite wave of two viscous shock waves for a one-dimensional system of non-viscous and heat-conductive ideal gas

TL;DR

This paper proves the asymptotic stability of a composite wave formed by two viscous shock waves in a one-dimensional heat-conductive ideal gas system without viscosity, extending previous results to a non-viscous case.

Contribution

It establishes the global existence and stability of a composite wave in a non-viscous, heat-conductive ideal gas system, a case not previously addressed.

Findings

Global solution exists and tends to the composite wave asymptotically.

Stability holds under small initial perturbations and wave strengths.

Unique determination of wave shifts by initial data.

Abstract

This paper is concerned with the asymptotic stability of a composite wave consisting of two viscous shock waves to the Cauchy problem for a one-dimensional system of heat-conductive ideal gas without viscosity. We extend the results by Huang-Matsumura \cite{Huang-Matsumura} where they treated the equation of viscous and heat-conductive ideal gas. That is, even forthe non-viscous and heat-conductive case, we show that if the strengths of the viscous shock waves and the initial perturbation are suitably small, the unique global solution in time exists and asymptotically tends toward the corresponding composite wave whose spacial shifts of two viscous shock waves are uniquely determined by the initial perturbation.