Instantaneous blowup and non-uniqueness of smooth solutions of MHD

TL;DR

This paper constructs smooth solutions to the incompressible MHD system that blow up instantaneously at the critical rate, using a novel convex integration scheme and a coupled geometric lemma.

Contribution

It introduces a new convex integration method for MHD that preserves the principal solution ansatz and a coupled geometric lemma of independent interest.

Findings

Solutions blow up instantaneously at the critical rate

The construction uses an inverse energy cascade mechanism

A new coupled geometric lemma is developed

Abstract

We construct a family of solutions $(u,B)$ of the incompressible magnetohydrodynamic (MHD) system, the $L^\infty$ norm of which blows up instantaneously at the critical rate. The solutions remain smooth except at the blowup time. An inverse energy cascade mechanism and a convex integration scheme along a time sequence are the main ingredients of the construction, inspired by our recent work [CDP25] for the Navier-Stokes equations. The challenge of the construction for the MHD system stems from the coupling and the necessity of preserving the same ansatz of the principal solution at every iterative step while implementing convex integration. Existing convex integration schemes for MHD can treat the coupling but fail to produce the same ansatz of the principal solution recursively. To achieve the goal, we introduce a coupled geometric lemma that decomposes a symmetric tensor and a skew-symmetric tensor simultaneously. We emphasize that such coupled geometric lemma is new and of independent interest.