Sharp decay characterization for the compressible Navier-Stokes equations

The low-frequency $L^1$ assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible Navier-Stokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical $L^p$ framework. Precisely, it is proved that the Besov space ${\dot{B}}_{2,\infty }^{{\sigma }_1}$-boundedness condition (with $\frac{d}{2}-\frac{2d}{p}\le {\sigma }_1<\frac{d}{2}-1$) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of ${\dot{B}}_{2,\infty }^{{\sigma }_1}$. To the best of our knowledge, our work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids.