Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions

Abstract

This work is concerned with the large time behavior of solutions to the barotropic compressible Navier-Stokes equations in ${\mathbb{R}}^d(d\ge 2)$. Precisely, it is shown that if the initial density and velocity additionally belong to some Besov space ${\dot{B}}_{2,\infty }^{-{\sigma }_1}$ with ${\sigma }_1\in (1-d/2,2d/p-d/2]$, then the $L^p$ norm (the slightly stronger ${\dot{B}}_{p,1}^0$ norm in fact) of global solutions admits the optimal decay $t^{-\frac{d}{2}(\frac{1}{2}-\frac{1}{p})-\frac{{\sigma }_1}{2}}$ for $t\to +\infty$. In contrast to refined time-weighted approaches ([11,43]), a pure energy argument (independent of the spectral analysis) has been developed in more general $L^p$ critical framework, which allows to remove the smallness of low frequencies of initial data. Indeed, bounding the evolution of ${\dot{B}}_{2,\infty }^{-{\sigma }_1}$-norm restricted in low frequencies is the key ingredient, whose proof mainly depends on non standard $L^p$ product estimates with respect to different Sobolev embeddings. The result can hold true in case of large highly oscillating initial velocities.

arXiv comment26 pages

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