An optimal regularity criterion for the 3D MHD equations in homogeneous Besov spaces

In this paper, we establish a new regularity criterion for weak solutions to the 3D incompressible MHD equations in terms of two pairs of(partial derivative(i)u(i), partial derivative(i)b(i)) (i=1,2,3). More precisely, it is proved that the weak solution (u, b) is smooth on (0, T], provided that for somei, j is an element of {1, 2, 3}with i not equal j, it holds that partial derivative(i)partial derivative(u), partial derivative(j)u(j), partial derivative(i)b(i), partial derivative(j)b(j) is an element of L-p(0, T; (B) over dot(q,infinity)(0)(R-3), 2/p + 3/q = 2, 3 <= q <= infinity.

Automated summary

In this paper, a new regularity criterion for weak solutions to the 3D incompressible MHD equations is established involving products of partial derivatives and conditions in the Lp space. It is proven that the solution is smooth over a certain time interval by satisfying certain criteria in the Lp space.