Global well-posedness of 3-D density-dependent incompressible MHD equations with variable resistivity

TL;DR

This work establishes global well-posedness for the 3-D inhomogeneous incompressible MHD system with density-dependent viscosity $\\mu(\\rho)$ and resistivity $\\sigma(\\rho)$, near unit strength in $L^{\\infty}$ and with density bounded away from vacuum. The authors develop a robust a priori framework in critical spaces using Littlewood-Paley theory, handling both variable and constant viscosity regimes, and construct global weak solutions via mollification and compactness. In the constant-viscosity case, they also prove uniqueness by a Lagrangian approach, under additional Besov regularity on the initial data. These results extend the theory of density-dependent Navier–Stokes to MHD and provide precise control in critical function spaces, with potential applications to density-patch problems in magnetized flows.

Abstract

In this paper, we investigate the global existence of weak solutions to 3-D inhomogeneous incompressible MHD equations with variable viscosity and resistivity, which is sufficiently close to $1$ in $L^\infty(\mathbb{R}^3),$ provided that the initial density is bounded from above and below by positive constants, and both the initial velocity and magnetic field are small enough in the critical space $\dot{H}^{\frac{1}{2}}(\mathbb{R}^3).$ Furthermore, if we assume in addition that the kinematic viscosity equals $1,$ and both the initial velocity and magnetic field belong to $\dot{B}^{\frac{1}{2}}_{2,1}(\mathbb{R}^3),$ we can also prove the uniqueness of such solution.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.2