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

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) - 3/4^{(2/p - 1)} when the initial data is large and belongs to H²(R³) ∩L^p(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) - 3/2 (1/p = 1/2) ^-1/2. 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)^-3/4 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)^-3/4 although the associated initial data is large.