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}.
Categories: math.AP