Abstract:

In this paper, the Cauchy problem of the MHD equations with damping is studied. When $\beta \ge 1$ and initial data satisfy ${u_0}$, ${b_0} \in {L^2}({{\mathbb{R} ^3}})$, the Galerkin method is used to prove the global weak solution of the equations. When the initial data satisfy ${u_0} \in H_0^1 \cap {L^{\beta + 1}}({{\mathbb{R} ^3}})$, ${b_0} \in H_0^1({{\mathbb{R} ^3}})$, it is possible to obtain a unique local strong solution for the equation group.

Key words: MHD equations, Damping, Weak solutions, Strong solutions