PreprintCC BY 4.0arXiv
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]×ℝn, n≥2, with a nonzero constant initial magnetic field B0 and arbitrary divergence-free velocity data v0∈ Hs, in the range (n+1)/2<s≤ 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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BibTeX
@misc{giardi2026ideal,
title = {{Ideal MHD below the classical well-posedness threshold}},
author = {Matteo Giardi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01093},
eprint = {2609.01093},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01093},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ideal MHD below the classical well-posedness threshold
AU - Matteo Giardi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01093
UR - https://arxiv.org/abs/2609.01093
AB - We establish local existence and uniqueness of solutions for the ideal incompressible magnetohydrodynamics system posed on [0,T]×ℝ^n, n≥2, with a nonzero constant initial magnetic field B_0 and arbitrary divergence-free velocity data v_0∈ H^s, in the range (n+1)/2<s≤ 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.
ER -