Optimal time-decay estimates for the compressible navier-stokes   equations in the critical l p framework
  The global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed in [7] more than fifteen years ago. However, whether (optimal) time-decay rates could be shown in general critical spaces and any dimension d $\ge$ 2 has remained an open question. Here we give a positive answer to that issue not only in the L 2 critical framework of [7] but also in the more general L p critical framework of [3, 6, 14]. More precisely, we show that under a mild additional decay assumption that is satisfied if the low frequencies of the initial data are in e.g. L p/2 (R d), the L p norm (the slightly stronger $\dot B^0_{p,1}$ norm in fact) 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 Sobolev regularity. Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. 
Rapha\"el Danchin, Jiang Xu
math.AP