Global nonlinear stability in three dimensions

Determine whether the global nonlinear stability result for the two-dimensional plasma-vacuum free boundary problem for ideal incompressible magnetohydrodynamics, with a strong magnetic field in the plasma region and zero magnetic field in the vacuum region, extends to the general three-dimensional case.

Background

The paper establishes global nonlinear stability for a two-dimensional plasma-vacuum free boundary problem governed by ideal incompressible magnetohydrodynamics in the plasma and trivial magnetic fields in the vacuum. The stability mechanism relies on a strong horizontal background magnetic field and on the propagation and interaction of Alfvén waves.

The authors explicitly leave unresolved whether an analogous global nonlinear stability theorem holds in the general three-dimensional setting. The issue is connected to the geometric and stability difficulties that arise when the plasma magnetic field is strong but the vacuum magnetic field is zero, a configuration that may fail the three-dimensional non-collinearity condition.

References

The global solutions obtained in this paper work in the two dimensional setting. Whether the global nonlinear stability result holds for the general three dimensional case is not clear.

— 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