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

TL;DR

This paper determines the optimal decay rates of solutions to the compressible Navier-Stokes equations with large initial data, establishing precise decay rates for solutions and their derivatives in various norms.

Contribution

It provides the first rigorous proof of the optimal decay rates for solutions and their derivatives in the large initial data setting for CNS equations.

Findings

Decay rate of the solution in $H^1$-norm is $(1+t)^{rac{3}{4}(rac{2}{p}-1)}$.

Decay rate of the first spatial derivative in $H^1$-norm is $(1+t)^{-rac{3}{2}(rac{1}{p}-rac{1}{2})-rac{1}{2}}$.

Lower bound of decay rate in $L^2$-norm is $(1+t)^{-rac{3}{4}}$ for $p=1$.

Abstract

In recent paper 5, it is shown that the upper decay rate of global solution of compressible Navier-Stokes(CNS) equations converging to constant equilibrium state $(1, 0)$ in $H^1-$norm is $(1+t)^{\frac34(\frac{2}{p}-1)}$ when the initial data is large and belongs to $H^2(\mathbb{R}^3) \cap L^p(\mathbb{R}^3) (p\in[1,2))$. Thus, the first result in this paper is devoted to showing that the upper decay rate of the first order spatial derivative converging to zero in $H^1-$norm is $(1+t)^{-\frac32(\frac1p-\frac12)-\frac12}$. For the case of $p=1$, the lower bound of decay rate for the global solution of CNS equations converging to constant equilibrium state $(1, 0)$ in $L^2-$norm is $(1+t)^{-\frac{3}{4}}$ if the initial data satisfies some low frequency assumption additionally. In other words, the optimal decay rate for the global solution of CNS equations converging to constant equilibrium state in $L^2-$norm is $(1+t)^{-\frac{3}{4}}$ although the associated initial data is large.