Onsager's Conjecture for Ideal Magnetohydrodynamics
Abstract
For any 0≤γ<1/3 we construct weak solutions (v,B,p) of the ideal MHD equations with v,B∈ Cγ(𝕋3×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.
Related
- Energy conservation of weak solutions and Onsager's conjecture for the two-component Camassa-Holm system
- The L3-based strong Onsager theorem
- The Numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion
- A stochastic convex integration scheme for the intermittent Onsager theorem in two-dimensional domains
- Droplets obstruct Onsager-type solutions of the Ising model in a magnetic field
BibTeX
@misc{giardi2026onsagers,
title = {{Onsager's Conjecture for Ideal Magnetohydrodynamics}},
author = {Matteo Giardi and László Székelyhidi Jr},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19506},
eprint = {2609.19506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19506},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Onsager's Conjecture for Ideal Magnetohydrodynamics
AU - Matteo Giardi
AU - László Székelyhidi Jr
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19506
UR - https://arxiv.org/abs/2609.19506
AB - For any 0≤γ<1/3 we construct weak solutions (v,B,p) of the ideal MHD equations with v,B∈ C^γ(𝕋^3×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.
ER -