Remarks on global regularity for the 3D MHD system with damping

We investigate the Cauchy problem for the 3D MHD system with damping terms |u|α−1u and |b|β−1b (α, β ≥ 1), and show that the strong solution exists globally and uniquely if one of the following four conditions holds, (1) 3≤α≤278,β≥4; (2) 278<α≤72,β≥72α−5; (3) 72<α<4,β≥5α+72α; (4) α ≥ 4, β ≥ 1. This improves the previous results significantly.