Splash-squeeze singularities and analytic breakdown in ideal incompressible MHD

TL;DR

This paper constructs a novel type of singularity in ideal incompressible MHD where the interface self-intersects without Sobolev norm blow-up, highlighting a new form of analytic breakdown driven by magnetic field behavior.

Contribution

It presents the first rigorous example of a squeeze-type singularity in free-boundary incompressible MHD, demonstrating analytic breakdown without Sobolev norm divergence.

Findings

Sobolev norms remain bounded at the singularity

Analyticity is lost at the self-intersection point

Magnetic field flattens to infinite order at the intersection

Abstract

We construct splash-squeeze singularities for the free boundary ideal incompressible plasma-vacuum system, in which two arcs of the plasma boundary come together to form a smooth, glancing self-intersection. As the interface self-intersects, Sobolev norms remain bounded, although analyticity is necessarily lost. This contrasts classical splash singularities, in which solutions remain analytic up to the time of self-intersection. The narrowing gap bounded by these arcs is not occupied by plasma, as squeezing the plasma itself would cause blow-up in Sobolev norms. Instead, the gap represents the region outside the plasma, a vacuum carrying a nontrivial magnetic field. The plasma on either side pinches the field as the gap closes, and, in response, the field flattens to infinite order at the intersection point (and nowhere else), thereby forming an analytic singularity. This gives the first example of analytic breakdown without Sobolev blow-up in a locally well-posed free-boundary incompressible fluid system, and can be viewed as the first rigorous construction of a squeeze-type singularity, in which we study and quantify the precise behavior of an active, incompressible vector field as it is completely pinched off by a free-boundary in finite time. The proof combines a magnetically-aligned Lagrangian formulation of ideal MHD together with weighted elliptic estimates in the vacuum that remain uniform as the width of the gap tends to zero. Our framework may provide a starting point for the analysis of squeeze-type singularities in other incompressible fluid models.