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In this paper, we prove the non-uniqueness of three-dimensionalmagneto-hydrodynamic (MHD) system in $C([0,T];L^2(\mathbb{T}^3))$ for anyinitial data in $H^{\bar{\beta}}(\mathbb{T}^3)$~($\bar{\beta}>0$), byexhibiting that the total energy and the cross helicity can be controlled in agiven positive time interval. Our results extend the non-uniqueness results ofthe ideal MHD system to the viscous and resistive MHD system. Different fromthe ideal MHD system, the dissipative effect in the viscous and resistive MHDsystem prevents the nonlinear term from balancing the stress error$(\RR_q,\MM_q)$ as doing in \cite{2Beekie}. We introduce the box flows andconstruct the perturbation consisting in six different kinds of flows in convexintegral scheme, which ensures that the iteration works and yields thenon-uniqueness.

On the weak solutions for the MHD systems with controllable total energy and cross helicity