Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

Abstract

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by $\kappa >0$ whose solution converges to that of the MHD equations as $\kappa \to 0$. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.

65 pages. Final version, accepted by J. Math. Pures Appl

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• The Magnetic Rayleigh-Taylor Instability in Three Dimensions
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Cited by in corpus (1)

• A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface