Sharp non-uniqueness of weak solutions to 3D magnetohydrodynamic equations

TL;DR

This paper demonstrates the non-uniqueness of weak solutions to 3D magnetohydrodynamic equations in certain supercritical regimes, revealing the critical role of the Ladyženskaja-Prodi-Serrin condition and showing solutions with partial regularity and failure of Taylor's conjecture.

Contribution

It establishes sharp non-uniqueness results for 3D MHD equations in supercritical regimes and links these to the Ladyženskaja-Prodi-Serrin condition, including partial regularity and vanishing viscosity limits.

Findings

Non-uniqueness of weak solutions in supercritical regimes.

Sharpness of non-uniqueness near the LPS endpoint.

Existence of solutions with partial regularity and fractal singular sets.

Abstract

We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class $L^\gamma_tW^{s,p}_x$, where the exponents $(s,\gamma,p)$ lie in two supercritical regimes. The result reveals that the scaling-invariant Lady\v{z}enskaja-Prodi-Serrin (LPS) condition is the right criterion to detect non-uniqueness, even in the highly viscous and resistive regime beyond the Lions exponent. In particular, for the classical viscous and resistive MHD, the non-uniqueness is sharp near the endpoint $(0,2,\infty)$ of the LPS condition. Moreover, the constructed weak solutions admit the partial regularity outside a small fractal singular set in time with zero $\mathcal{H}^{\eta_*}$-Hausdorff dimension, where $\eta_*$ can be any given small positive constant. Furthermore, we prove the strong vanishing viscosity and resistivity result, which yields the failure of Taylor's conjecture along some subsequence of weak solutions to the hyper viscous and resistive MHD beyond the Lions exponent.