Global stability of large solutions to the 3D compressible Navier-Stokes equations

TL;DR

This paper proves the global stability of large solutions to the 3D compressible Navier-Stokes equations, demonstrating convergence to equilibrium and constructing large solutions with initial data in critical spaces.

Contribution

It introduces a new mechanism for convergence, proves global stability for large solutions, and constructs solutions with arbitrarily large initial velocities in critical spaces.

Findings

Solutions converge to equilibrium with heat-equation-like decay rates

Global-in-time stability for solutions close initially

Existence of large solutions with arbitrary initial vertical velocity

Abstract

The present paper aims at the investigation of the global stability of large solutions to the compressible Navier-Stokes equations in the whole space. Our main results and innovations can be concluded as follows: Under the assumption that the density $\rho(t,x)$ verifies $\rho(0,x)\ge c>0$ and $\sup_{t\ge0}\|\rho(t)\|_{C^\alpha}\le M$ with $\alpha$ sufficiently small, we establish a new mechanism for the convergence of the solution to its associated equilibrium with an explicit decay rate which is as the same as that for the heat equation. The main idea of the proof relies on the basic energy identity, techniques from blow-up criterion and a new estimate for the low frequency part of the solution. We prove the global-in-time stability for the equations, i.e, any perturbed solution will remain close to the reference solution if initially they are close to each other. Our result implies that the set of the smooth and bounded solutions is an open set. Going beyond the close-to-equilibrium setting, we construct the global large solutions to the equations with a class of initial data in $L^p$ type critical spaces. Here the "large solution" means that the vertical component of the velocity could be arbitrarily large initially.