Basing on the Newton's second law and the Maxwell equations for the electromagnetic fields, we establish a new 3D incompressible magneto-hydrodynamics(MHD) equations for the motion of plasma under the standard Coulomb gauge. By using the Galerkin method, we prove a global weak solution for this 3D new model. keywords Maxwell equations, Plasma, Coulomb gauge, Magneto-hydrodynamics, Galerkin method. * Email:liuruikuan2008@163.com. Supported by NSFC(11401479) † Corresponding author:jiayan 1985@163.com; and obtained partial regularity theorems. Kang and Lee [10] and Wang and Zhang [22] gave other regularity criteria under the Ladyzhenskaya-Prodi-Serrin type conditions. Kang and Kim [11], [12] considered suitable weak solutions in the half space and gave a boundary regularity criteria.Recently, the Hall magneto-hydrodynamic (Hall-MHD) model was established by A. Marion, D. Pierre etc in [14]. The Hall-MHD model receives an increasing attentions from plasma physicists. It is believed to be the key for understanding the problem of magnetic reconnection. Indeed, space plasma observations provide strong evidence for the existence of frequent and fast changes in the topology of magnetic field lines, associating to important events such as solar flares [6]. For the existence of global weak and regular solutions of the (Hall-MHD) equations, we refer to [7], [8], [13]. For the generalized magnetohydrodynamics (MHD) and Navier-Stokes systems, we refer to [1], [6], [13], [17], [18], [19], [23], [24].However, the corresponding models in above papers are obtain by using certain approximations and assumptions. In this paper, basing on some basic physics principles, a new 3D incompressible magneto-hydrodynamic model for plasma shall be established without any assumptions. It is in particular that the model describes the un-static electronic field. Moreover, the existence of the global weak solution of this 3D model is obtained by the Galerkin method.
