Global well-posedness for the resistive MHD equations in critical Besov spaces

Weikui Ye, Zhaoyang Yin

Published 2021-05-07Version 1

In this paper, we mainly investigate the Cauchy problem of the resistive MHD equation without viscosity. 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 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),~s>2$ when the initial data satisfies $\int_{\mathbb{S}^2}b_0dx=0$ and $\|b_0\|_{L^2(\mathbb{S}^2)}+\|u_0\|_{L^2(\mathbb{S}^2)}\leq \epsilon$. 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}.