Global existence and optimal time-decay rates of the compressible Navier-Stokes equations with density-dependent viscosities

Abstract: This paper is devoted to studying the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosities given by $\mu=\rho^\alpha,\lambda=\rho^\alpha(\alpha>0)$. We establish the global existence and optimal decay rates of classical solutions under the assumptions of small initial data in $L^1(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ and the viscosity constraint $|\alpha-1|\ll 1$. The key idea of our proof lies in the combination of Green's function method, energy method and a time-decay regularity criterion. In contrast to previous works, the Sobolev norms of the spatial derivatives of the initial data may be arbitrarily large in our analysis