Sharp non-uniqueness of weak solutions to 3D magnetohydrodynamic equations: Beyond the Lions exponent

We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class L gamma t Ws,p viscosity and resistivity can be larger than the Lions exponent 5/4 and (s, gamma, p) lies in two supercritical regimes with respect to the Lady & zcaron;enskaja-Prodi-Serrin (LPS) condition. The constructed weak solutions admit the partial regularity outside a small fractal singular set in time with zero H eta & lowast;Hausdorff dimension, with eta & lowast; being any given small positive constant. In particular, for the canonical viscous and resistive MHD, the non-uniqueness is sharp near one endpoint of the LPS condition, which extends the recent result in [22] for Navier-Stokes equations. The partial regularity for MHD equations is also new. Furthermore, the strong vanishing viscosity and resistivity result is obtained, it yields the failure of Taylor's conjecture along some sequence of weak solutions to the hyper viscous and resistive MHD equations. Our proof utilizes the spatial-temporal intermittent convex integration scheme, the temporal building blocks feature the almost optimal intermittency, which improves the recent ones constructed in [58]. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.