A Regularity Result for the Incompressible Magnetohydrodynamics Equations with Free Surface Boundary

We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and that of the magnetic field. To the best of our knowledge this is the first result that focuses on the low regularity solution for incompressible free-boundary MHD equations.

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