Decay of Strong Solution for the Compressible Navier-Stokes Equations with Large Initial Data

TL;DR

This paper proves the convergence and decay rates of large solutions to the compressible Navier-Stokes equations in three dimensions, extending previous decay results and establishing explicit algebraic decay rates for solutions with initial data in negative Sobolev spaces.

Contribution

It establishes the preservation of negative Sobolev norms and derives explicit decay rates for large solutions, including the decay of spatial derivatives, in the context of compressible Navier-Stokes equations.

Findings

Negative Sobolev norms are preserved over time.

Global solutions decay algebraically to equilibrium.

Decay rate of spatial derivatives matches heat equation rate.

Abstract

In this paper, we investigate the convergence of the global large solution to its associated constant equilibrium state with an explicit decay rate for the compressible Navier-Stokes equations in three-dimensional whole space. Suppose the initial data belongs to some negative Sobolev space instead of Lebesgue space, we not only prove the negative Sobolev norms of the solution being preserved along time evolution, but also obtain the convergence of the global large solution to its associated constant equilibrium state with algebra decay rate. Besides, we shall show that the decay rate of the first order spatial derivative of large solution of the full compressible Navier-Stokes equations converging to zero in $L^2-$norm is $(1+t)^{-5/4}$, which coincides with the heat equation. This extends the previous decay rate $(1+t)^{-3/4}$ obtained in \cite{he-huang-wang2}.