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On the Free Boundary Problems for the Ideal Incompressible MHD Equations
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We investigate the general plasma-vacuum interface problems for the ideal incompressible MHD equations with or without surface tension and prove their nonlinear local well-posedness in standard Sobolev spaces under either non-zero surface tension or the stability condition that the magnetic fields are everywhere non-collinear on the interface. In particular, the results show that both capillary forces and tangential magnetic fields can stabilize the motion of the plasma-vacuum interfaces. Moreover, the vanishing surface tension limit results are established under the Rayleigh-Taylor sign condition or the non-collinearity condition. All these results hold with no graph assumption on the free interface.
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Sharp well-posedness for the free boundary MHD equations
Free boundary incompressible magnetohydrodynamics is shown to be well-posed at the sharp low-regularity scale s > d/2 + 1, with first uniqueness and smooth-solution construction results in a new state space.
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