| Abstract: |
| Firstly, we consider the conditional regularity of mild solution $v$ to the incompressible Navier-Stokes equations in three dimensions. Let $e \in \mathbb{S}^2$ and $0 < T^\ast < \infty$.
J. Chemin and P. Zhang (Ann. Sci. \`{E}c. Norm. Sup\`{e}r, 2016) proved the regularity of $v$ on $(0,T^\ast]$ if there exists $p$ belongs to (4, 6) such that
$$\int_0^{T^\ast}\|v\cdot e\|^p_{\dot{H}^{\frac{1}{2}+\frac{2}{p}}}dt < \infty.$$
J. Chemin, P. Zhang and Z. F. Zhang (Arch. Ration. Mech. Ana. ,2017) extended the range of $p$ to $(4, \infty)$. In this talk we settle the case p belongs to [2, 4] and apply this method to the 3D incompressible MHD system. |
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