Global large strong solutions to the compressible Navier-Stokes equations with density-dependent viscosities, case I: isentropic flows

TL;DR

This paper proves the global existence of strong solutions for 3D compressible Navier-Stokes equations with density-dependent viscosities, even for large initial data, contrasting with classical small-perturbation results.

Contribution

It establishes the existence of global strong solutions for large initial data in the density-dependent viscosity case, using an elliptic-dominated structure and energy functional analysis.

Findings

Global strong solutions exist for large initial data.

The system's elliptic structure is key to the analysis.

Results extend classical small-perturbation theories.

Abstract

In this paper, we consider the Cauchy problem for the three-dimensional barotropic compressible Navier-Stokes equations with density-dependent viscosities. By considering the system as an elliptic-dominated structure and defining suitable energy functionals, after the elaborate index analysis, we establish the global existence of strong solutions as long as the initial data is large enough. This is a big contrast to the classical results where the initial data is a small perturbation of some resting states.