Time-asymptotic stability of generic Riemann solutions for compressible Navier-Stokes-Fourier equations

TL;DR

This paper proves the long-term stability of complex wave patterns in the compressible Navier-Stokes-Fourier equations, demonstrating convergence of solutions to a combination of shock, contact, and rarefaction waves over time.

Contribution

It is the first to establish the time-asymptotic stability of generic Riemann solutions involving multiple wave types for these equations.

Findings

Solutions converge uniformly to a combination of waves as time approaches infinity.

The method of a-contraction with shifts effectively handles the complex wave interactions.

The approach addresses a longstanding open problem in the field.

Abstract

We establish the time-asymptotic stability of solutions to the one-dimensional compressible Navier-Stokes-Fourier equations, with initial data perturbed from Riemann data that forms a generic Riemann solution. The Riemann solution under consideration is composed of a viscous shock, a viscous contact wave, and a rarefaction wave. We prove that the perturbed solution of Navier-Stokes-Fourier converges, uniformly in space as time goes to infinity, to a viscous ansatz composed of viscous shock with time-dependent shift, a viscous contact wave and an inviscid rarefaction wave. This is a first resolution of the challenging open problem associated with the generic Riemann solution. Our approach relies on the method of a-contraction with shifts, specifically applied to both the shock wave and the contact discontinuity wave. It enables the application of a global energy method for the generic combination of three waves.