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∈L21−r(0,T,Ẋr(R3)) with r∈[0,1[r∈[0,1[, then the solution remains smooth on [0,T][0,T]. Since L3r(R3)⊂Ẋr(R3) for 0≤r<32, our result improve the results in He and Xin (2005) [1] and Zhou (2005) [2].