Onsager's Conjecture for Ideal Magnetohydrodynamics

Abstract

For any $0\le \gamma <1/3$ we construct weak solutions $(v,B,p)$ of the ideal MHD equations with $v,B\in C^{\gamma }({\mathbb{T}}^3\times \mathbb{R})$, which conserve neither the total energy nor the cross-helicity and have nontrivial magnetic helicity. We also establish anisotropic Hölder bounds with distinct velocity and magnetic exponents and stronger regularity along every magnetic field line. The admissible exponents connect the Goldreich--Sridhar \cite{GoldreichSridhar1995} spatial and parallel pair $1/3,1/2$ with the $1/4$ spatial scaling of the Iroshnikov--Kraichnan weak-turbulence picture \cite{Iroshnikov1963,Kraichnan1965}. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is regarded 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 result of Enciso, Peñafiel-Tomás and Peralta-Salas \cite{EnPePe} and can be used to reprove \cite{GiRa} of Giri and Radu without a Newton iteration.

AI-assisted technical extension of our $C^{1/5-}$ result and the first author's PhD Thesis; see AI declaration for details. The text overlap in the introduction of the two papers is intentional