Local stability in the three-dimensional zero-vacuum-field setting

Establish whether local-in-time stability holds for the three-dimensional plasma-vacuum free boundary problem for ideal incompressible magnetohydrodynamics when the plasma contains a strong magnetic field and the vacuum magnetic field is zero.

Background

The paper treats the two-dimensional plasma-vacuum free boundary problem with a strong horizontal background magnetic field in the plasma and a trivial magnetic field in the vacuum. It proves global nonlinear stability in that two-dimensional configuration and sketches local well-posedness.

The corresponding three-dimensional local-in-time stability problem remains unresolved according to the authors. They note that the zero-vacuum-field configuration appears not to satisfy the non-collinearity stability condition used in three-dimensional plasma-vacuum interface theory.

References

Under strong magnetic field in the plasma region and zero magnetic field in the vacuum region, the local in time stability result is not clear for the corresponding three dimensional problem.

— Global nonlinear stability of the plasma-vacuum free boundary problem for the 2D ideal incompressible MHD  (2609.10009 - Cai, 9 Sep 2026) in Remark immediately following the main theorem in Section 1, Introduction