We consider a free boundary problem for the incompressible ideal magnetohydrodynamic equations that describes the motion of the plasma in vacuum. The magnetic field is tangent and the total pressure vanishes along the plasma-vacuum interface. Under the Taylor sign condition of the total pressure on the free surface, we prove the local well-posedness of the problem in Sobolev spaces.
Construction of GCM hypersurfaces in perturbations of Kerr
A construction of blow up solutions for co-rotational wave maps
Velocity blow-up in $C^1\cap H^2$ for the 2D Euler equations
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