The Estimate of Lifespan and Local Well‐Posedness for the Nonresistive MHD Equations in Homogeneous Besov Spaces

ABSTRACT In this paper, we mainly investigate the Cauchy problem of the nonresistive MHD equation. We first establish the local existence in the homogeneous Besov space with , and give a lifespan of the solution that depends on the norm of the Littlewood–Paley decomposition of the initial data. Then, we prove that if the initial data in , then the corresponding existence times , 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 . Therefore, the nonresistive MHD equation is local well‐posedness in the homogeneous Besov space in the Hadamard sense. Our obtained result improves considerably the recent results by Fefferman et al[Arch. Ration. Mech. Anal. 223, no. 7 (2017):677‐691], Chemin et al [Adv. Math. 286(2016):1‐31 ] and Li et al[Adv. Math. 317(2017): 786‐798].