Global Fujita-Kato solutions of the incompressible inhomogeneous magnetohydrodynamic equations

TL;DR

This work establishes global well-posedness for the 3D incompressible inhomogeneous MHD equations with bounded and possibly discontinuous density in the critical-regularity setting. It develops a robust framework based on critical Besov and Lorentz spaces, maximal regularity, and dyadic decompositions to handle rough density and nonlinear couplings between velocity and magnetic field, achieving both global existence and decay, as well as a general uniqueness result without strong regularity on the density. The authors prove three main results: global existence with small density variation and small data in $\dot{B}^{-1+3/p}_{p,1}$ for $1<p<3$ (and without the variation condition for $p=2$), a unique global Fujita–Kato solution with velocity/magnetic data small in $\dot{B}^{1/2}_{2,\infty}$ but possibly large in $\dot{H}^{1/2}$, and a broad uniqueness principle for bounded, nonnegative density. These contributions extend the global solvability theory to inhomogeneous MHD with rough density and provide detailed temporal decay and stability estimates with potential applications to geophysical and plasma contexts.

Abstract

We investigate the incompressible inhomogeneous magnetohydrodynamic equations in $\mathbb{R}^3$, under the assumptions that the initial density $ρ_0$ is only bounded, and the initial velocity $u_0$ and magnetic field $B_0$ exhibit critical regularities. In particular, the density is allowed to be piecewise constant with jumps. First, we establish the global-in-time well-posedness and large-time behavior of solutions to the Cauchy problem in the case that $ρ_0$ has small variations, and $u_0$ and $B_0$ are sufficiently small in the critical Besov space $\dot{B}^{3/p-1}_{p,1}$ with $1<p<3$. Moreover, the small variation assumption on $ρ_0$ is no longer required in the case $p=2$. Then, we construct a unique global Fujita-Kato solution under the weaker condition that $u_0$ and $B_0$ are small in $\dot{B}^{1/2}_{2,\infty}$ but may be large in $\dot{H}^{1/2}$. Additionally, we show a general uniqueness result with only bounded and nonnegative density, without assuming the $L^1(0,T;L^{\infty})$ regularity of the velocity. Our study systematically addresses the global solvability of the inhomogeneous magnetohydrodynamic equations with rough density in the critical regularity setting.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 3.1
• proof