Sharp and strong non-uniqueness for the magneto-hydrodynamic equations

Abstract

In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the three-dimensional magneto-hydrodynamic (MHD) system. More precisely, we show that any weak solution $(v,b)\in L_t^p{L}_x^{\infty }$ is non-unique in $L_t^p{L}_x^{\infty }$ with $1\le p<2$, which reveals the strong non-uniqueness, and the sharpness in terms of the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint $(2,\infty )$. Moreover, for any $1\le p<2$ and $\epsilon >0$, we construct non-Leray-Hopf weak solutions in $L_t^p{L}_x^{\infty }\cap L_t^1{C}^{1-\epsilon }$. The results of Navier-Stokes equations in \cite{1Cheskidov} imply the sharp non-uniqueness of MHD system with trivial magnetic field $b$. Our result shows the non-uniqueness for any weak solution $(v,b)$ including non-trivial magnetic field $b$.