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In this paper, we mainly investigate the Cauchy problem of the non-resistiveMHD equation. We first establish the local existence in the homogeneous Besovspace $\dot{B}^{\frac{d}{p}-1}_{p,1}\times \dot{B}^{\frac{d}{p}}_{p,1}$ with$p<\infty$, and give a lifespan $T$ of the solution which depends on the normof the Littlewood-Paley decomposition of the initial data. Then, we prove thatif the initial data $(u^n_0,b^n_0)\rightarrow (u_0,b_0)$ in$\dot{B}^{\frac{d}{p}-1}_{p,1}\times \dot{B}^{\frac{d}{p}}_{p,1}$, then thecorresponding existence times $T_n\rightarrow T$, which implies that they havea common lower bound of the lifespan. Finally, we prove that thedata-to-solutions map depends continuously on the initial data when $p\leq 2d$.Therefore the non-resistive MHD equation is local well-posedness in thehomogeneous Besov space in the Hadamard sense. Our obtained result improvesconsiderably the recent results in \cite{Li1,chemin1,Feffer2}.

The estimate of lifespan and local well-posedness for the non-resistive MHD equations in homogeneous Besov spaces