Global Fujita-Kato solutions of the incompressible inhomogeneous magnetohydrodynamic equations

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 ${\dot{B}}_{p,1}^{3/p-1}$ 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 ${\dot{B}}_{2,\infty }^{1/2}$ 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.

50pages