Remarks on the blow-up criterion for the MHD system involving horizontal components or their horizontal gradients

Regularity Criteria for the 3D Magneto-Hydrodynamics Equations in Anisotropic Lorentz Spaces

“…Recently, many works have been done for mixed-norm spaces. Stefanov-Torres [28] obtained the boundedness of Calderón-Zygmund operators on mixed-norm Lebesgue spaces. Georgiadis et al [29] obtained various properties of anisotropic Triebel-Lizorkin spaces with mixed norms.…”

“…Stefanov-Torres [28] obtained the boundedness of Calderón-Zygmund operators on mixed-norm Lebesgue spaces. Georgiadis et al [29] obtained various properties of anisotropic Triebel-Lizorkin spaces with mixed norms. In [30], Chen-Sun introduced the iterated weak and weak mixed-norm spaces and given some applications to geometric inequalities.…”

“…where i = 1, 2, 3, then the solution is smooth on interval (0,T]. For readers interested in this topic for partial components, please refer to [20][21][22][23][24][25][26] for recent progresses. Motivated by papers cited above, the aim of this article is to study the regularity of weak solutions for the 3D MHD equations (1) in term of the two partial derivative of the velocity components and magnetic components on framework of the anisotropic Lorentz space.…”

“…Motivated by papers cited above, the aim of this article is to study the regularity of weak solutions for the 3D MHD equations (1) in term of the two partial derivative of the velocity components and magnetic components on framework of the anisotropic Lorentz space. Before stating our main Theorem, we shall first recall the definitions of some function spaces [27].…”

Regularity criteria for the 3D magnetohydrodynamics system involving only two velocity components

“…[14][15][16] Note that the conditions (2) and 3are optimal from the scaling-invariant point of view, and formally, they can also imply the regularity of the Navier-Stokes equations. However, because the flow is incompressible, one may expect regularity conditions in terms of partial components of velocity field, then there have been many criteria concerning partial components for the MHD equations [17][18][19][20] and for the Navier-Stokes equations. [21][22][23][24][25][26][27][28][29][30] Among these results, one found that the coupling effect of magnetic field cannot be neglected for these regularity conditions.…”