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

TL;DR

This paper proves the global existence and optimal decay rates of solutions to the 3D compressible Navier-Stokes equations with density-dependent viscosities, allowing large initial derivatives under small initial data assumptions.

Contribution

It establishes global solutions and decay rates for density-dependent viscosities with minimal restrictions on initial data derivatives, using a novel combination of analytical methods.

Findings

Global existence of classical solutions

Optimal decay rates established

Large initial derivatives allowed

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