Time-asymptotic stability of composite wave for the one-dimensional compressible fluid of Kortwewg type

TL;DR

This paper proves the asymptotic stability of a combined wave pattern in the one-dimensional Korteweg-type compressible fluid system, demonstrating convergence of solutions to a composite wave under small initial perturbations.

Contribution

It introduces a novel application of the $a$-contraction method with shift to establish stability of composite waves in the NSK system.

Findings

Solutions converge to the composite wave asymptotically.

Stability holds under small initial perturbations.

Method successfully applied to hyperbolic systems with dispersive effects.

Abstract

We study the asymptotic stability of a composition of rarefaction and shock waves for the one-dimensional barotropic compressible fluid of Korteweg type, called the Navier-Stokes-Korteweg(NSK) system. Precisely, we show that the solution to the NSK system asymptotically converges to the composition of the rarefaction wave and shifted viscous-dispersive shock wave, under certain smallness assumption on the initial perturbation and strength of the waves. Our method is based on the method of $a$-contraction with shift developed by Kang and Vasseur \cite{KV16}, successfully applied to obtain contraction or stability of nonlinear waves for hyperbolic systems.