The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

Abstract

In a recent paper (He et al., 2019), it is shown that the upper decay rate of global solution of compressible Navier–Stokes(CNS) equations converging to constant equilibrium state (1,0) in H1−norm is (1+t)−34(2p−1) when the initial data is large and belongs to H2(R3)∩Lp(R3)(p∈[1,2)). Thus, the first result in this paper is devoted to showing that the upper decay rate of the first order spatial derivative converging to zero in H1−norm is (1+t)−32(1p−12)−12. For the case of p=1, the lower bound of decay rate for the global solution of CNS equations converging to constant equilibrium state (1,0) in L2−norm is (1+t)−34 if the initial data satisfies some low frequency assumption additionally. In other words, the optimal decay rate for the global solution of CNS equations converging to constant equilibrium state in L2−norm is (1+t)−34 although the associated initial data is large.