The Mhd Equations In The Space L ~ ¡Þ (0, T; ~ N (¦¸)) In The Energy Equation And The Uniqueness Of Weak Solutions

We consider the initial value problem of the following MHD equations:Here Ω is a n-dimensional domain in R~n, not necessarily bounded, u =denote the unknown velocity fields, magnetic fields, the scalar function of pressure respectively. f = f(x,t) = (f~1(x,t),...f~n(x,t)) is a given external force. a, b denote the given initial data satisfying ▽? a = 0, ▽? b = 0.In this paper, we are mainly concerned with the energy equality and the uniqueness of the weak solutions of the MHD equations (*1) in L~∞(0, T; L~n(Ω)). The contents of the paper include two parts:1. We consider the energy equality of the weak solutions of the MHD equations (*1). To do so, we first prove that both u(t) and B(t) are weakly continuous in the norm of L_σ~2. Then,by this property, we obtain the energy equality satisfied by the solution to (*1).2. We consider the uniqueness of weak solutions. we employ the energy equality to prove the uniqueness of the weak solutions of (*1) in L∞(0, T; L~n(Ω)) under the assumptions that u(t) and B(t) is right continuous for all t ∈[0, T) in the norm of L~n(Ω). Furthermore, we remove the assumptions on the right continuity of u(t) and B(t) in the norm of L~n(Ω) by constructing a strong solutions of the approximation equations of (*1).