Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system

Abstract

We study the partial regularity problem for a three dimensional simplified Ericksen–Leslie system, which consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier–Stokes equations, we first prove some new local energy bounds and an $ \varepsilon $-regularity criterion for suitable weak solutions to the simplified Ericksen-Leslie system, i.e., for $ \sigma\in [0,1] $, there exists a $ \varepsilon>0 $ such that if $ (u,d,P) $ is a suitable weak solution in $ Q_{r}(z_{0}) $ with $ 0<r\leq 1 $ and $ z_{0} = (x_{0},t_{0}) $, and satisfies \begin{document}$ \begin{align*} r^{-\frac{3}{2-\sigma}}\!\!\int_{\!t_{0}-r^{2}}^{t_{0}}\! (\||u|^{2}\|_{\!H^{-\sigma}(B_{r}(x_{0}))}^{\frac{2}{2-\sigma}} \!+\!\||\nabla d|^{2}\|_{\!H^{-\sigma}(B_{r}(x_{0}))}^{\frac{2}{2-\sigma}} \!+\!\|P\|_{\!H^{-\sigma}(B_{r}(x_{0}))}^{\frac{2}{2-\sigma}})\text{d}t\leq \varepsilon, \end{align*} $\end{document} then $ (u, d) $ is regular at $ z_{0} $. Here, $ H^{-\sigma}(B_{r}(x)) $ is the dual space of $ H^{\sigma}_{0}(B_{r}(x)) $, the space of functions $ f $ in the homogeneous Sobolev space $ \dot{H}^{\sigma}(\mathbb{R}^{3}) $ such that $ \operatorname{supp} f\subset \overline{B_{r}(x)} $. Inspired by this $ \varepsilon $-regularity criterion, we then improve the known upper Minkowski dimension of the possible interior singular points for suitable weak solutions from $ \frac{95}{63} (\approx 1.50794) $ given by [24] (Nonlinear Anal. RWA, 44 (2018), 246–259.) to $ \frac{835}{613} (\approx 1.36215) $.