Ideal MHD below the classical well-posedness threshold

Abstract

We establish local existence and uniqueness of solutions for the ideal incompressible magnetohydrodynamics system posed on $[0,T]\times {\mathbb{R}}^n$, $n\ge 2$, with a nonzero constant initial magnetic field ${\mathbf{B}}_0$ and arbitrary divergence-free velocity data $v_0\in H^s$, in the range $(n+1)/2<s\le n/2+1$. The proof uses a Lagrangian wave--Hodge reformulation and exploits an Alfvén null--structure hidden in the pressure forcing. In particular, the constructed Eulerian solutions are induced by a bi-Lipschitz measure-preserving flow map. Zhang first identified this null-structure in \cite{Zhang2024}; the present work provides a self-contained bridge from that Lagrangian theory to the Eulerian Cauchy problem.

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