On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in ${\mathbb{R}}^3$

Abstract

The purpose of this article is twofold: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in ${\mathbb{R}}^3$ with initial data $(u_0,b_0)$ of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference $u_0-b_0$. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field $b$ under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.

30 pages