$C^{1/5^{-}}$ Convex Integration Solutions of Ideal MHD

TL;DR

This paper constructs weak solutions to the ideal MHD equations with Hölder regularity below 1/5 that do not conserve key physical quantities, extending recent convex integration methods.

Contribution

It introduces new convex integration solutions for ideal MHD with regularity below 1/5, not conserving energy or cross-helicity, and links geometric constructions to Lie algebra techniques.

Findings

Constructed solutions in $C^eta$ for $eta<1/5$ that violate conservation laws.

Extended convex integration methods to the setting of ideal MHD.

Connected geometric Lie algebra techniques with fluid dynamics solutions.

Abstract

For any $0\leq \gamma < 1/5$, we construct weak solutions $(v, B, p )$ of the Ideal MHD Equations which do not conserve the total kinetic energy, the cross-helicity and lie in $C^\gamma(\mathbb{T}^3\times\mathbb{R})$. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is thought of as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent work of and building on the recent work of Enciso, Pe\~nafiel-Tom\'as and Peralta-Salas.