  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.

Analysis of PDEs