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

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