On the free boundary problem of incompressible ideal MHD
Abstract：In this talk, I will review some results of free boundary problem of incompressible ideal MHD in a bounded domain based on the joint works with T. Luo. We raised a Taylor-sign type condition for MHD and obtained the a priori estimates for the nonlinear system. We also proved the ill-posedness of the system in 2D disk domain if it did not satisfy this Taylor-sign type condition on the free boundary. We also studied the well-posedness for the linearized system. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute 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 linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain 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.