Splash-squeeze singularities and analytic breakdown in ideal incompressible MHD

TL;DR

The paper constructs splash-squeeze singularities for the free-boundary ideal incompressible MHD system, where a plasma interface pinch occurs while the exterior vacuum magnetic field remains nontrivial. It achieves this via a magnetically aligned Lagrangian formulation that yields a solvable wave system for surface and interior fields, complemented by δ-weighted elliptic estimates in pinch domains to control vacuum effects. A backward-in-time construction from smooth data demonstrates the formation of a splash, while a forward analytic-data argument shows that analyticity must fail at the pinch, establishing analytic breakdown without Sobolev blow-up. The results rely on a delicate Dirichlet-to-Neumann cancellation in the vacuum, weighted estimates, and a carefully designed iterative scheme built on Kato's theory to solve the resulting quasilinear wave system.

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.

Figures (9)

• Figure 1: Before a splash
• Figure 2: Splash
• Figure 3: Plasma chamber before splash
• Figure 4: Plasma chamber at splash time
• Figure 5:

Theorems & Definitions (296)

• Theorem 1.1
• Remark 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 2.6
• proof : Proof of Proposition \ref{['JacobianIdent']}