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

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 $(1+t{)}^{\frac{3}{4}(\frac{2}{p}-1)}$ 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 $(1+t{)}^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}}$. 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.

18 pages