On the 2D plasma-vacuum interface problems for ideal incompressible MHD

This manuscript concerns the stability conditions for the well-posedness of the two-dimensional plasma-vacuum interface problems for ideal incompressible magnetohydrodynamics (MHD) equations, which describe the dynamics of conducting perfect fluids in a vacuum region under the influence of magnetic fields. Due to the counterexamples constructed by C. Hao and the second author [ Comm. Math. Phys. 376 (2020), 259-286], the plasma-vacuum problems violating the Rayleigh-Taylor sign condition and without surface tension are believed to be highly unstable or even ill-posed. However, based on a geometric perspective, we demonstrate that, although the initial data provided therein are ill-posed in the Lagrangian framework characterizing the regularity of flow maps, plasma-vacuum problems with such initial data can still be stable/well-posed in the Eulerian setting without involving flow maps, under the hypothesis that the total magnetic fields are non-degenerate on the free boundary. This discloses the essential difference between the Eulerian and Lagrangian frameworks of MHD free boundary problems and the stabilizing effect of the magnetic field. We also consider the surface tension effect and establish the local well-posedness theories in standard Sobolev spaces in this case. Furthermore, under either the non-degeneracy assumption on magnetic fields or the Rayleigh-Taylor sign condition on the effective pressure, we prove vanishing surface tension limits. These results indicate that both capillary forces and non-degenerate tangential magnetic fields can stabilize the motion of the plasma-vacuum interface, and the stabilizing effect of surface tension is stronger than that of the tangential magnetic fields. © 2025 Elsevier Inc.