... mass is conserved. Moreover, if we multiply 1.3, 1.4 ,and 1.5 by u, w,andb, respectively, and sum up the resulting equations, we have by using 1.2 that12ρu2w212b2t12ρuu2w2 ... usual Sobolev space, and Wm,2Ω HmΩ, W0,pΩ LpΩ.For simplicity, we denote by C the various generic positive constants depending only on thedata and T, and use the following ... the magnetic fields b and b thatb −bt ub − bxu − ubxu − uxb uxb − b − w − wx υb − bxx, 4.9which, multiplied by 2b −b and integrated by parts over Ω,givesddtb...