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

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 $T_nrightarrow T$, 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 $pleq 2d$. 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}.