In this talk, we will give a sharp decay characterization of the Cauchy problem for the compressible Navier-Stokes equations in the critical regularity framework. Precisely, we consider the Besov space Bs2,∞ including the case s=d/2 associated with the embedding in L1. We prove that the Bs2,∞ boundedness for the low-frequency part of initial perturbation is not only sufficient but also necessary to achieve upper bounds of time-decay estimates of solutions. Furthermore, we establish upper and lower bounds of time-decay estimates if and only if the low-frequency part of the initial perturbation belongs to a nontrivial subset of Bs2,∞.
