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 (u₀, b₀) in critical Besov spaces $B_{p,1}^{{d \over p} + 1} \times B_{p,1}^{{d \over p}}$ with 1 ≤ p ≤ ∞, and give a lifespan T of the solution which depends on the norm of the Littlewood—Paley decomposition (profile) 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 $C\left( {[0,\infty } \right);{H^s}\left. {\left( {{\mathbb{S}^2}} \right)} \right) \times \left( {C\left( {[0,\infty } \right);{H^{s - 1}}\left. {\left( {{\mathbb{S}^2}} \right)} \right) \cap {L^2}\left( {[0,\infty } \right);{H^s}\left. {\left( {{\mathbb{S}^2}}\right)} \right)} \right)$ with s> 2, when the initial data satisfies $\int_{{\mathbb{S}^2}} {{b_0}dx} = 0$ and ${\left\| {{u_0}} \right\|_{B_{\infty ,1}^1\left( {{\mathbb{S}^2}} \right)}} + {\left\| {{b_0}} \right\|_{B_{\infty ,1}^0\left( {{\mathbb{S}^2}} \right)}} \le \epsilon$. It’s worth noting that our results imply some large and low regularity initial data for the global existence.