A logarithmically improved regularity criterion for the 3D MHD equations in Morrey-Campanato space

Abstract

In this paper, we will establish a sufficient condition for the regularity criterion to the 3D MHD equation in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure $\partial _{3}\pi $ satisfies the logarithmical Serrin type condition $\partial _{3}\pi $ satisfies the logarithmical Serrin type condition \begin{equation*} \int_{0}^{T}\frac{\left\Vert \partial _{3}\pi (s)\right\Vert _{\overset{% \cdot }{\mathcal{M}}_{2,\frac{3}{r}}}^{\frac{2}{2-r}}}{1+\ln (1+\left\Vert b(s)\right\Vert _{L^{4}})}ds<\infty \text{ for }0<1, \end{equation*} then the solution $(u,b)$ remains smooth on $\left[ 0,T\right] $. Compared to the Navier-Stokes result, there is a logarithmic correction involving $b$ in the denominator.