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

Abstract

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 ${\dot{B}}_{p,1}^{\frac{d}{p}-1}\times {\dot{B}}_{p,1}^{\frac{d}{p}}$ 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 $(u_0^n,b_0^n)\to (u_0,b_0)$ in ${\dot{B}}_{p,1}^{\frac{d}{p}-1}\times {\dot{B}}_{p,1}^{\frac{d}{p}}$, then the corresponding existence times $T_n\to T$, 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 $p\le 2d$. 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}.

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