Sharp non-uniqueness of weak solutions to 2D magnetohydrodynamic equations

In this paper, we prove that weak solutions to the 2D viscous and resistive magnetohydrodynamic (MHD) equations are non-unique in $L_t^2{L}^p({\mathbb{R}}^2)\cap L_t^1{W}^{1,p}({\mathbb{R}}^2)$ for given any $1\le p<\mathrm{\infty }$, showing the sharpness of the Ladyzhenskaya–Prodi–Serrin condition at the endpoint $(2,\mathrm{\infty })$ and the solutions live on the borderline of the Beale–Kato–Majda criterion. To the best of our knowledge, this is the first non-uniqueness result for the 2D viscous and resistive MHD system. As byproducts, we also obtain non-uniqueness for the Navier–Stokes equations in $L_t^2{L}^p$ with $1\le p<\mathrm{\infty }$, and for the MHD system with large ${\mathrm{B}\mathrm{M}\mathrm{O}}^{-1}$ initial data.