Time-asymptotic behavior of composite waves of viscous shocks for compressible barotropic Navier-Stokes equations

Abstract

Over the past few decades, time-asymptotic stability of the solution to the Cauchy problem for the one- dimensional compressible barotropic Navier-Stokes equations has been studied in relation to the Riemann problem for the Euler equation. In this paper, we prove the time-asymptotic stability of composite wave of two viscous shock waves for the Navier-Stokes equations when the Riemann solution of the Euler equation is given as composition waves of two shock waves. The large time behavior of the Navier Stokes equation for composite waves of two viscous shock waves is left as the last problem after Kang, Vasseur, and Wang showed the large time behavior of Navier-Stokes equations for the composite waves of viscous shock waves and rarefaction waves in [7]. Our results solve the time asymptotic stability problem of compressible Navier-Stokes equations. Our method relies on a-contraction methods with shifts developed by Kang and Vasseur [4]. The biggest challenge in dealing with the composition of two shock waves is to control them as a single diffusion term. To overcome this difficulty, we introduce a localization function to successfully segment the diffusion term to control each shock wave and use its careful analysis.