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

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 $\mathrm{\mid }\alpha -1\mathrm{\mid }\ll 1$. 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