Serrin–type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component

In this paper, we consider the Cauchy problem to the 3D MHD equations. We show that the Serrin–type conditions imposed on one component of the velocity u3 and one component of magnetic fields b3 with u3 ∈ L 0(−1, 0;L0(B(2))), b3 ∈ L 1(−1, 0;L1(B(2))), 2 p0 + 3 q0 = 2 p1 + 3 q1 = 1 and 3 < q0, q1 < +∞ imply that the suitable weak solution is regular at (0, 0). The proof is based on the new local energy estimates introduced by Chae-Wolf (Arch. Ration. Mech. Anal. 2021) and Wang-Wu-Zhang (arXiv:2005.11906).

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