Buckmaster–Vicol Onsager-type conjecture for Elsässer energies in ideal MHD

Let $(u,b)$ be a weak solution of the ideal MHD system, and define the total energy and cross helicity by

$$\mathcal{E}(t)=\frac{1}{2}\int_{\mathbb{T}^d}(|u(x,t)|^2+|b(x,t)|^2)\,\mathrm{d}x,$$ $$\mathcal{H}_c(t)=\int_{\mathbb{T}^d}u\cdot b\,\mathrm{d}x.$$

Here $C_{x,t}^\alpha$ denotes joint Hölder regularity and $B_{3,\infty}^\alpha$ the corresponding Besov space. Buckmaster–Vicol's Onsager-type conjecture. (a) Any weak solution $(u,b)$ belonging to $C_{x,t}^\alpha$ or $L_t^3B_{3,\infty}^\alpha$ for $\alpha>1/3$ conserves $\mathcal{E}$ and $\mathcal{H}_c$. (b) For any $\alpha<1/3$, there exist weak solutions $(u,b)\in C_{x,t}^\alpha$ or $L_t^3B_{3,\infty}^\alpha$ that dissipate $\mathcal{E}$ and for which $\mathcal{H}_c$ is not constant in time. This conjecture seeks the critical regularity threshold for conservation of the total energy and cross helicity in ideal MHD. The rigid part has been resolved by standard arguments, while the flexible part is the subject of the stated conjecture.