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

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