Global Fujita-Kato solutions of the incompressible inhomogeneous magnetohydrodynamic equations

TL;DR

This paper establishes the global existence, uniqueness, and large-time behavior of solutions to the inhomogeneous incompressible magnetohydrodynamic equations with rough density and critical regularity initial data, including piecewise constant densities.

Contribution

It introduces new global well-posedness results for the MHD equations with minimal regularity assumptions on density and velocity fields, extending previous theories to rough density scenarios.

Findings

Proved global-in-time solutions for small initial data in critical Besov spaces.

Constructed unique solutions under weaker conditions on initial velocity and magnetic field.

Demonstrated uniqueness with only bounded, nonnegative density without additional regularity assumptions.

Abstract

We investigate the incompressible inhomogeneous magnetohydrodynamic equations in $\mathbb{R}^3$, under the assumptions that the initial density $\rho_0$ 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 $\rho_0$ 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 $\rho_0$ 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.