Remark on the regularity criterion for three-dimensional magnetohydrodynamic equations

In this letter, we consider the three-dimensional magnetohydrodynamic (3D MHD) equations and we prove a new regularity criterion for weak solutions. More precisely, we show that if the velocity field u ∈ L 2 1 − r ( 0 , T , X ̇ r ( R 3 ) ) with r ∈ [ 0 , 1 [ , then the solution remains smooth on [ 0 , T ] . Since L 3 r ( R 3 ) ⊂ X ̇ r ( R 3 ) for 0 ≤ r < 3 2 , our result improve the results in He and Xin (2005) [1] and Zhou (2005) [2].