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The global existence issue for the isentropic compressible Navier-Stokesequations in the critical regularity framework has been addressed in [7] morethan fifteen years ago. However, whether (optimal) time-decay rates could beshown in general critical spaces and any dimension d $\ge$ 2 has remained anopen question. Here we give a positive answer to that issue not only in the L 2critical framework of [7] but also in the more general L p critical frameworkof [3, 6, 14]. More precisely, we show that under a mild additional decayassumption that is satisfied if the low frequencies of the initial data are ine.g. L p/2 (R d), the L p norm (the slightly stronger $\dot B^0_{p,1}$ norm infact) of the critical global solutions decays like t --d(1 p -- 1 4) for t$\rightarrow$ +$\infty$, exactly as firstly observed by A. Matsumura and T.Nishida in [23] in the case p = 2 and d = 3, for solutions with high Sobolevregularity. Our method relies on refined time weighted inequalities in theFourier space, and is likely to be effective for other hyperbolic/parabolicsystems that are encountered in fluid mechanics or mathematical physics.

Optimal time-decay estimates for the compressible navier-stokes equations in the critical l p framework