A regularity criterion for the 3D generalized MHD system involving partial components

This paper is devoted to establishing the global regularity involving magnetic fields and partial components of the velocity for the 3D generalized magnetohydrodynamic equations with dissipation terms -(-Delta)alpha u and -(-Delta)(ss)b. We assume 1 <= a = ss <= 3/2 and prove that if b, u(3) epsilon L-w (0, T; L-q(R-3)) with 2 alpha/w + 3/q <= 3(2a-1)/4 + 3(1-epsilon)/4q, 3+epsilon/2 alpha-1 < q <= infinity, and 0 < epsilon <= 1/3, then the local strong solution is smooth on [0, T].