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In this paper, we mainly investigate the Cauchy problem of the non-viscousMHD equations with magnetic diffusion. We first establish the localwell-posedness (existence,~uniqueness and continuous dependence) with initialdata $(u_0,b_0)$ in critical Besov spaces ${B}^{\frac{d}{p}+1}_{p,1}\times{B}^{\frac{d}{p}}_{p,1}$ with $1\leqp\leq\infty$, and give a lifespan $T$ of the solution which depends on the normof the Littlewood-Paley decomposition of the initial data. Then, we prove theglobal existence in critical Besov spaces. In particular, the results of globalexistence also hold in Sobolev space $ C([0,\infty);{H}^{s}(\mathbb{S}^2))\times \Big(C([0,\infty);{H}^{s-1}(\mathbb{S}^2))\capL^2\big([0,\infty);{H}^{s}(\mathbb{S}^2)\big)\Big)$ with $s>2$, when theinitial 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 lowregularity initial data for the global existence, which improves considerablythe recent results in \cite{weishen}.

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