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Decay of Strong Solution for the Compressible Navier-Stokes Equations with Large Initial Data

In this paper, we investigate the convergence of the global large solution to its associated constant equilibrium state with an explicit decay rate for the compressible Navier-Stokes equations in three-dimensional whole space. Suppose the initial data belongs to some negative Sobolev space instead of Lebesgue space, we not only prove the negative Sobolev norms of the solution being preserved along time evolution, but also obtain the convergence of the global large solution to its associated constant equilibrium state with algebra decay rate. Besides, we shall show that the decay rate of the first order spatial derivative of large solution of the full compressible Navier-Stokes equations converging to zero in $L^2-$norm is $(1+t)^{-5/4}$, which coincides with the heat equation. This extends the previous decay rate $(1+t)^{-3/4}$ obtained in \cite{he-huang-wang2}.