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We establish the a priori estimates and prove a blow-up criterion for thethree-dimensional free boundary incompressible ideal magnetohydrodynamicsequations. The fluid occupies a bounded region with a free boundary that is aclosed surface, without assumptions of simple connectedness or periodicity ofthe region (thus, Fourier transforms cannot be applied), nor the graphassumption for the free boundary. The fluid is under the influence of surfacetension, and flattening the boundaries using local coordinates is insufficientto resolve this problem. This is because local coordinates fail to preservecurvature, as the mean curvature of a flat boundary degenerates to zero. Toaddress these challenges and circumvent the intricate issue of spatialregularity in Lagrangian coordinates, we utilize reference surfaces torepresent the free boundary and develop new energy functionals that bothpreserve the material derivative and incorporate spatial-temporal scaling$\partial_t \sim \nabla^{\frac{3}{2}}$ in Eulerian coordinates. This methodenables us to establish both low-order and high-order regularity estimateswithout any loss of regularity. More importantly, we prove a blow-up criterionand provide a complete classification of blow-ups, including theself-intersection of the free boundary (which the graph assumption cannothandle), the breakdown of the mean curvature, and the blow-up of the normalvelocity (which Lagrangian coordinates fail to capture). To the best of ourknowledge, this is the first result addressing the a priori estimates and theblow-up criterion for free boundary problems with surface tension in generalregions.

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