Some new regularity criterion for MHD three-dimensional flow

The aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in R².We show that if (∇_h^u, ∂₃,b₃) ∈ L^2α,2γ with 2/α+3/γ ≤3, 1≤γ ≤ ∞, then the weak solution is regular and unique; if (∇_h^u, ∇_hb) ∈L^2α,2γ with 2/α +3/γ ≤ 3, 1 ≤ γ ≤ ∞, then the weak solution is regular and unique; if (∂₃^u, ∇u ₃) ∈ L^2α,2γ and (u₃,b,∂₃b, ∇b₃) ∈ L^4α,4γ. with 2/α+3/γ ≤ 3, 1 ≤ γ ≤ ∞, then the weak solution is regular and unique. Here we use the notation ∇_h = (∂₁,∂₂).