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The Optimal Decay Rate of Strong Solution for the Compressible Navier-Stokes Equations with Large Initial Data (1911.04791v1)

Published 12 Nov 2019 in math.AP

Abstract: In 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 $H1-$norm is $(1+t){\frac34(\frac{2}{p}-1)}$ when the initial data is large and belongs to $H2(\mathbb{R}3) \cap Lp(\mathbb{R}3) (p\in[1,2))$. 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 $H1-$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 $L2-$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 $L2-$norm is $(1+t){-\frac{3}{4}}$ although the associated initial data is large.

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