On Onsager-type conjecture for the Els\"{a}sser energies of the ideal MHD equations

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 Els\"{a}sser energies of the ideal MHD equations. More precisely, for \(\beta < 1/3\), we construct weak solutions \((u, b) \in C^\beta([0,T] \times \mathbb{T}^3)\) 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 H\"{o}lder-continuous weak solutions with non-trivial magnetic fields. Specifically, for \(\beta < 1/5\), there exist infinitely many solutions \((u, b) \in C^\beta([0,T] \times \mathbb{T}^2)\) 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 H\"{o}lder-continuous Euler solution can be approximated by a sequence of $C^\beta$-weak solutions for the ideal MHD equations in the $L^p$-topology for $1\le p<\infty$.