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

In this paper, we investigate the ideal magnetohydrodynamics (MHD) equations on tours ${\text{TTT}}^d$. For $d=3$, we resolve the flexible part of Onsager-type conjecture for Elsässer energies of the ideal MHD equations. More precisely, for $\beta <1\mathrm{/}3$, 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 Hölder-continuous weak solutions with non-trivial magnetic fields. Specifically, for $\beta <1\mathrm{/}5$, 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 Hölder-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 $1\le p<\mathrm{\infty }$.