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

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 $B_{p,1}^{\frac{d}{p}+1}\times B_{p,1}^{\frac{d}{p}}$ with $1\le p\le \infty$, 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 $C([0,\infty );H^s({\mathbb{S}}^2))\times (C([0,\infty );H^{s-1}({\mathbb{S}}^2))\cap L^2([0,\infty );H^s({\mathbb{S}}^2)\left)\right)$ with $s>2$, when the initial data satisfies ${\int }_{{\mathbb{S}}^2}{b}_{0}dx=0$ and $\Vert u_0{\Vert }_{B_{\infty ,1}^1({\mathbb{S}}^2)}+\Vert b_0{\Vert }_{B_{\infty ,1}^0({\mathbb{S}}^2)}\le \epsilon$. 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}.

arXiv commentThe non-viscous MHD equations with magnetic diffusion, Local well-posedness, critical Besov spaces, global existence. arXiv admin note: text overlap with arXiv:2012.03489

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Cited by in corpus (1)

• Global Regularity and instability for the incompressible non-viscous Oldroyd-B model