  In this paper, we mainly investigate the Cauchy problem of the non-resistive
MHD equation. We first establish the local existence in the homogeneous Besov
space $\dot{B}^{\frac{d}{p}-1}_{p,1}\times \dot{B}^{\frac{d}{p}}_{p,1}$ with
$p<\infty$, and give a lifespan $T$ of the solution which depends on the norm
of the Littlewood-Paley decomposition of the initial data. Then, we prove that
if 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 the
corresponding existence times $T_n\rightarrow T$, which implies that they have
a common lower bound of the lifespan. Finally, we prove that the
data-to-solutions map depends continuously on the initial data when $p\leq 2d$.
Therefore the non-resistive MHD equation is local well-posedness in the
homogeneous Besov space in the Hadamard sense. Our obtained result improves
considerably the recent results in \cite{Li1,chemin1,Feffer2}.

Analysis of PDEs