Some new regularity criterion for MHD three-dimensional flow - Rahman, Saeed ur; Hayat, Tasawar; Alsulami, Hamed H. (Kuwait Journal of Science, 2017)
The aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in R^{3}. We show that if (grad_{h}u,\partial_{3}b_{3}) belongs to L^{2alpha,2gamma} with 2/alpha+3/gamma<=3, 1<=gamma<=infinity, then the weak solution is regular and unique; if (grad_{h}u, grad_{h}b) belongs to L^{2alpha,2gamma} with 2/alpha+3/gamma}<=3, 1<=gamma<=infinity, then the weak solution is regular and unique; if (partial_{3}u, grad u_{3}) belongs to L^{2alpha,2gamma} and (u_{3}, b, \partial_{3} b, grad b_{3}) belongsL^{4alpha,4gamma} with 2/alpha+3/gamma}<=3, 1<=gamma<=infinity, then the weak solution is regular and unique. Here we use the notation grad_{h}=(partial_{1}, partial_{2}).