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arXiv · 2602.06979

Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values

Baishun Lai·Ge Tang·Ziying Xu
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Abstract

This paper is concerned with the weak solution theory for the MHD system with large $L^3$-initial data. Due to the fact that the natural boundary condition on the magnetic field $H$ is the slip boundary condition, the Leray-Schauder fixed-point theorem, which have used to investigate the weak solution theory of the Navier-Stokes system, becomes invalid. To address such difficulty, we will invoke the Leray's approximation technique and the perturbation theory to seek a global weak solution to the Cauchy problem for MHD equations with large $L^3$-initial data. Our strategy provides a simple alternative (self-contained) proof of weak $L^3$-solution theory of incompressible Navier-Stokes system. Moreover, this weak solution is unique under some restrictions.

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Baishun Lai, Ge Tang, Ziying Xu. 2026-01-24. Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values. https://arxiv.org/abs/2602.06979

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