Stability of large solutions for full compressible Navier-Stokes equations in the whole spaces

TL;DR

This paper proves the global-in-time stability and convergence to equilibrium of large solutions to the full compressible Navier-Stokes equations in the whole space, under certain boundedness conditions on density and temperature.

Contribution

It establishes the stability and convergence rates of large solutions to the full Navier-Stokes-Fourier system in unbounded space, extending understanding of solution behavior.

Findings

Solutions converge to equilibrium at heat equation rate.

Propagation of positive lower bounds for density and temperature.

Stability of solutions under small perturbations.

Abstract

The current paper is devoted to the investigation of the global-in-time stability of large solutions for the full Navier-Stokes-Fourier system in the whole space. Suppose that the density and the temperature are bounded from above uniformly in time in the Holder space $C^\alpha$ with $\alpha$ sufficiently small and in $L^\infty$ space respectively. Then we prove two results: (1). Such kind of the solution will converge to its associated equilibrium with a rate which is the same as that for the heat equation if we impose the same condition on the initial data. As a result, we obtain the propagation of positive lower bounds of the density and the temperature. (2). Such kind of the solution is stable, that is, any perturbed solution will remain close to the reference solution if initially they are close to each other. This shows that the set of the smooth and bounded solutions is open.