A Beale–Kato–Majda criterion for free boundary incompressible ideal magnetohydrodynamics

Abstract

We prove a continuation criterion for the free boundary problem of three-dimensional incompressible ideal magnetohydrodynamic (MHD) equations in a bounded domain, which is analogous to the theorem given in Beale, Kato, and Majda [Commun. Math. Phys. 94, 61–66 (1984)]. We combine the energy estimates of our previous works [C. Hao and T. Luo, Arch. Ration. Mech. Anal. 212(3), 805–847 (2014)] on incompressible ideal MHD and some analogous estimates in Ginsberg [SIAM J. Math. Anal. 53, 3366–3384 (2021); arXiv:1811.06154] to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds.