Global well-posedness for the non-viscous MHD equations with magnetic diffusion in critical Besov spaces

TL;DR

This paper proves the local and global well-posedness of the non-viscous MHD equations with magnetic diffusion in critical Besov spaces, allowing for large and low regularity initial data, thus advancing the mathematical understanding of these equations.

Contribution

It establishes the first global existence results for the non-viscous MHD equations in critical Besov spaces with large and low regularity initial data.

Findings

Local well-posedness in critical Besov spaces

Global existence under certain initial conditions

Improves upon recent results by handling larger and less regular data

Abstract

In this paper, we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion. We first establish the local well-posedness (existence,~uniqueness and continuous dependence) with initial data $(u_0,b_0)$ in critical Besov spaces $ {B}^{\frac{d}{p}+1}_{p,1}\times{B}^{\frac{d}{p}}_{p,1}$ with $1\leq p\leq\infty$, and give a lifespan $T$ of the solution which depends on the norm of the Littlewood-Paley decomposition of the initial data. Then, we prove the global existence in critical Besov spaces. In particular, the results of global existence also hold in Sobolev space $ C([0,\infty); {H}^{s}(\mathbb{S}^2))\times \Big(C([0,\infty);{H}^{s-1}(\mathbb{S}^2))\cap L^2\big([0,\infty);{H}^{s}(\mathbb{S}^2)\big)\Big)$ with $s>2$, when the initial data satisfies $\int_{\mathbb{S}^2}b_0dx=0$ and $\|u_0\|_{B^1_{\infty,1}(\mathbb{S}^2)}+\|b_0\|_{{B}^{0}_{\infty,1}(\mathbb{S}^2)}\leq \epsilon$. It's worth noting that our results imply some large and low regularity initial data for the global existence, which improves considerably the recent results in \cite{weishen}.