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 ˙ 2 , ∞ - σ 1 (R d) with the regularity σ 1 ∈ (1 - d / 2 , 2 d / 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. [ABSTRACT FROM AUTHOR]