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

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 $(v,b)\in L^p_tL^{\infty}_x$ is non-unique in $L^p_tL^{\infty}_x$ 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 $ε>0$, we construct non-Leray-Hopf weak solutions in $L^p_tL^{\infty}_x\cap L^1_tC^{1-ε}$. 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 $b$.