Global well-posedness for the 3D compressible Navier-Stokes equations in optimal Besov space

Abstract

We consider the Cauchy problem to the 3D barotropic compressible Navier-Stokes equation. We prove global well-posedness, assuming that the initial data $({\rho }_0-1,u_0)$ has small norms in the critical Besov space ${\mathbb{X}}_p={\dot{B}}_{p,1}^{3/p}({\mathbb{R}}^3)\times {\dot{B}}_{p,1}^{-1+3/p}({\mathbb{R}}^3)$ for $2\le p<6$ and $({\rho }_0-1,{\rho }_0{u}_0)$ satisfies an additional low frequency condition. Our results extend the previous results in \cite{FD2010, CMZ2010, H20112} where $p<4$ is needed for high frequency, to the optimal range $p<6$. The main ingredients of the proof consist of: a novel nonlinear transform that uses momentum formulation for low-frequency and effective velocity method for high frequency, and estimate of parabolic-dispersive semigroup that enables a $L^q$-framework for low frequency.