Large-time behavior of composite waves of viscous shocks for the barotropic Navier-Stokes equations

TL;DR

This paper proves that the 1D barotropic Navier-Stokes flow, perturbed from Riemann data, converges over time to a composition of two viscous shock waves with independently small amplitudes, demonstrating a novel large-time behavior result.

Contribution

It is the first to establish convergence of the flow to a composite wave of two viscous shocks with independently small strengths.

Findings

Flow converges to a composite of two viscous shocks over time

Convergence is uniform in space and includes dynamical shifts

Waves' strengths can be chosen independently

Abstract

We study the large-time behavior of the 1D barotropic Navier-Stokes flow perturbed from Riemann data generating a composition of two shock waves with small amplitudes. We prove that the perturbed Navier-Stokes flow converges, uniformly in space, towards a composition of two viscous shock waves as time goes to infinity, up to dynamical shifts. Especially, the strengths of the two waves can be chosen independently. This is the first result for the convergence to a composite wave of two viscous shocks with independently small amplitudes.