Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations

TL;DR

This paper establishes the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes and MHD equations in critical Besov spaces without small density variation assumptions, extending previous results to lower initial velocity regularity.

Contribution

It introduces a new a priori estimate for elliptic equations with nonconstant coefficients in Besov spaces, enabling well-posedness results at lower regularity levels.

Findings

Proves global well-posedness for 3D inhomogeneous Navier-Stokes equations in critical Besov spaces.

Extends the well-posedness results to inhomogeneous incompressible MHD equations.

Develops a novel a priori estimate for elliptic equations with variable coefficients in Besov spaces.

Abstract

The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier-Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (Arch. Ration. Mech. Anal. 204 (1):189--230, 2012, and J. Math. Pures Appl. 100 (1):166--203, 2013) to a more lower regularity index about the initial velocity. The key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in Besov spaces which have the same degree as $L^2$ in $\mathbb{R}^3$. Finally, we also generalize our well-posedness result to the inhomogeneous incompressible MHD equations.