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Abstract
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The global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed by R. Danchin more than fifteen years ago. However, whether (optimal) time-decay rates could be shown in general critical spaces and any dimension $d\geq2$ has remained an open question. Here we give a positive answer to that issue not only in the $L^2$ critical framework but also in the more general $L^p$ critical framework, which is exactly as firstly observed by A. Matsumura and T. Nishida in the case $p=2$ and $d=3,$ for solutions with high Sobolev regularity. This is a joint work with R. Danchin.
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