In this talk, I review some results of free boundary problem of incompressible ideal MHD in a bounded domain with closed free surfaces based on the joint works with Prof. T. Luo, especially the well-posedness for the linearized system. We expressed the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substituted the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearized this equation with respect to the position vector field whose time derivative was the velocity, and obtained the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.

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