On Onsager-type conjecture for the Elsässer energies of the ideal MHD equations

TL;DR

This work establishes the flexible portion of the Onsager-type conjecture for the Elsässer energies in the ideal MHD system. In 3D on $\mathbb{T}^3$, it shows that for any $0<\beta<\tfrac{1}{3}$ there are infinitely many weak solutions $(u,b)\in C^\beta$ that dissipate energy and fail to conserve cross helicity, achieved via a symmetry reduction to a 2D Euler-with-tracer system and a Newton–Nash convex integration scheme. It further proves an energy-dissipative variant with non-conserved cross helicity, using a precise energy profile $e(t)$. In 2D on $\mathbb{T}^2$, for $0<\beta<\tfrac{1}{5}$ there are infinitely many weak solutions with nontrivial magnetic fields and energy dissipation, obtained via a spatial-separation convex integration strategy that decouples magnetic effects from velocity perturbations and leverages energy-corrector constructions. As a byproduct, any Hölder Euler solution can be approximated in $L^p$ by $C^\beta$-weak MHD solutions, linking MHD dynamics with Euler irregularity. Overall, the results confirm the 1/3 threshold for conservation of the total energy and cross helicity in 3D, while demonstrating sharp flexible behavior in both 3D and 2D through sophisticated convex integration methods that exploit symmetry and spatial separation.

Abstract

In this paper, we investigate the ideal magnetohydrodynamics (MHD) equations on tours $\TTT^d$. For $d=3$, we resolve the flexible part of Onsager-type conjecture for Elsässer energies of the ideal MHD equations. More precisely, for \(β< 1/3\), we construct weak solutions \((u, b) \in C^β([0,T] \times \mathbb{T}^3)\) with both the total energy dissipation and failure of cross helicity conservation. The key idea of the proof relies on a symmetry reduction that embeds the ideal MHD system into a 2$\frac{1}{2}$D Euler flow and the Newton-Nash iteration technique recently developed in \cite{GR}. For $d=2$, we show the non-uniqueness of Hölder-continuous weak solutions with non-trivial magnetic fields. Specifically, for \(β< 1/5\), there exist infinitely many solutions \((u, b) \in C^β([0,T] \times \mathbb{T}^2)\) with the same initial data while satisfying the total energy dissipation with non-vanishing velocity and magnetic fields. The new ingredient is developing a spatial-separation-driven iterative scheme that incorporates the magnetic field as a controlled perturbation within the convex integration framework for the velocity field, thereby providing sufficient oscillatory freedom for Nash-type perturbations in the 2D setting. As a byproduct, we prove that any Hölder-continuous Euler solution can be approximated by a sequence of $C^β$-weak solutions for the ideal MHD equations in the $L^p$-topology for $1\le p<\infty$.

Theorems & Definitions (60)

• Definition 1.1: Weak solution
• Conjecture 1.1: Onsager-type conjecture for the Elsässer energies BV21
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Proposition 2.1