Time-Asymptotic Stability of Generic Riemann Solutions for Compressible Navier–Stokes–Fourier Equations

We establish the time-asymptotic stability of generic Riemann solutions to the one-dimensional compressible Navier–Stokes–Fourier equations. The Riemann solution under consideration is a generic combination of a shock, a contact discontinuity, and a rarefaction wave. We prove that the perturbed solution of Navier–Stokes–Fourier converges, uniformly in space as time goes to infinity, to an 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 time-asymptotic stability of three waves of different kinds associated with the generic Riemann solutions. Our approach relies on the method of a-contraction with shifts and relative entropy, specifically applied to both the shock wave and the contact wave. It enables the application of a global energy method for the generic combination of three waves.