Abstract:

In this paper, we consider the regularity of weak solutions to the incompressible NS equations and MHD equations in the Triebel-Lizorkin space and multiplier space respectively. By using Littlewood-Paley decomposition and energy estimate methods, we proved that if horizontal velocity ũ=(u₁, u₂, 0) satisfies

$ \nabla_{h}\tilde{u}\in L^{p}(0,T; \dot{F}^{0}_{q,\frac{2q}{3}}(\mathbb{R} ^{3})), ~~~~~\frac{2}{p}+\frac{3}{q}=2,~~~~\frac{3}{2}<q\leq\infty, $

then the weak solution is actually the unique strong solution on[0, T). For MHD equations, we prove that if horizontal velocity and magnetic field satisfies

$ (\tilde{u},\tilde{b})\in L^{\frac{2}{1-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[0, 1), $

or horizontal gradient satisfies

$(\nabla_{h}\tilde{u},\nabla_{h}\tilde{b})\in L^{\frac{2}{2-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[0, 1], $

then the weak solution is actually unique strong solution on[0, T).

Key words: NS equations, MHD equations, Regularity criteria