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

Abstract

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 $1\le p<\infty$, showing the sharpness of the Ladyzhenskaya--Prodi--Serrin condition at the endpoint $(2,\infty )$ 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 $1\le p<\infty$, and for the MHD system with large ${\mathrm{BMO}}^{-1}$ initial data.