A Low-Frequency Assumption for Optimal Time-Decay Estimates to the Compressible Navier-Stokes Equations

The global existence issue in critical spaces for compressible Navier-Stokes equations, was addressed by Danchin (Invent Math 141:579-614, 2000) and then developed by Charve and Danchin (Arch Rational Mech Anal 198:233-271, 2010), Chen et al. (Commun Pure Appl Math 63:1173-1224, 2010) and Haspot (Arch Rational Mech Anal 202:427-460, 2011) in more general L-p setting. The main aim of this paper is to exhibit (more precisely) time-decay estimates of solutions constructed in the critical regularity framework. To the best of our knowledge, the low-frequency assumption usually plays a key role in the large-time asymptotics of solutions, which was firstly observed by Matsumura and Nishida (J Math Kyoto Univ 20:67-104, 1980) in the L-1(R-d) space. We now claim a new low-frequency assumption for barotropic compressible Navier-Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov space <(B)over dot>(-sigma 1)(2,infinity)(R-d) with the regularity sigma(1) is an element of(1-d/2,2d/p-d/2], then a sharp time-weighted inequality including enough time-decay information can be available, where optimal decay exponents for the high frequencies are exhibited. The proof mainly depends on some non standard Besov product estimates. As a by-product, those optimal time-decay rates of L-q-L-r type are also captured in the critical framework.