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

Abstract

This paper is dedicated to the regularity criteria of weak solutions to the three‐dimensional incompressible magnetohydrodynamics system in terms of only two velocity components. It is proved that the weak solution (u,b) is smooth on (0,T], provided that ∇u1,∇u2∈L43−2sfalse(0,T;trueB˙∞,∞−sfalse(ℝ3false)false).3emwith.3em0<s<1, or u1,u2∈L4false(0,T;BMOfalse(ℝ3false)false).

Regularity via one vorticity component for the 3D axisymmetric MHD equations

“…Precisely, they showed if the velocity field satisfies 𝑢 ∈ 𝐿 then the weak solution of (1.1) is smooth on ℝ 3 × (0, 𝑇]. Then, this result was extended or improved by [10,43,44,51].…”

“…Precisely, they showed if the velocity field satisfies $$$$\begin{equation} u\in L^{\alpha ,\gamma }\text{ with }\frac{2}{\alpha }+\frac{3}{\gamma }\le 1,\text{ \ }3&lt;\gamma \le \infty , \end{equation}$$$$or $$$$\begin{equation} \nabla u\in L^{\alpha ,\gamma }\text{ with }\frac{2}{\alpha }+\frac{3}{ \gamma }\le 2,\text{ \ }\frac{3}{2}&lt;\gamma \le \infty , \end{equation}$$$$then the weak solution of (1.1) is smooth on $$\mathbb {R}^{3}\times (0,T]$$. Then, this result was extended or improved by [10, 43, 44, 51]. Global regularity can also be obtained by imposing conditions on the pressure [21, 40, 58].…”

An optimal regularity criterion for the 3D MHD equations in homogeneous Besov spaces

“…Over the past decade, more and more researchers are interested in investigating the regularity criteria for the MHD equations in terms of partial components or partial derivatives of the velocity field, the magnetic field, or the pressure, see, for example, the literature 28‐39 . Ji‐Lee 37 obtained a regularity condition in terms of horizontal components of the velocity field and the magnetic field, that is, where .…”