Number of Singular Points and Energy Equality for the Co-rotational Beris–Edwards System Modeling Nematic Liquid Crystal Flow

Abstract

In this paper, we consider the singular points of suitable weak solutions to the 3D co-rotational Beris–Edwards system modeling the hydrodynamical motion of nematic liquid crystal flows, which is a coupled system with the Navier–Stokes equations for the fluid and a parabolic system of Q-tensor for the liquid average orientation. We prove that if \((\textbf{u},Q)\) defined on \(\mathbb {R}^{3}\times (0,T)\) is a suitable weak solution to the 3D co-rotational Beris–Edwards system, and satisfies

$$\begin{aligned} \Vert (\textbf{u},\nabla Q)\Vert _{L^{q,\infty }(0,T;L^{p}(\mathbb {R}^{3}))}<\infty \text { with }3<p<\infty \text { and } \frac{2}{q}+\frac{3}{p}=1, \end{aligned}$$

then for a given open subset \(\Omega \subseteq \mathbb {R}^{3}\) and for a given moment of time \(t_0\in (0,T)\), the number of points of the set \(\Sigma (t_0)\cap \Omega \) is finite, where \(\Sigma (t_0)\equiv \{(x,t_0)\in \Sigma \}\) and \(\Sigma \) is the set of singular points for \((\textbf{u},Q)\). Moreover, if \(T_{1}\in (0,T)\) is the first time for singularity appears, and if \((\textbf{u},Q)\) satisfies

$$\begin{aligned} \Vert (\textbf{u},\nabla Q)(\cdot ,t)\Vert _{L^p(\mathbb {R}^3)} \le \frac{c_0}{(T_1-t)^{\frac{p-3}{2p}}} \quad \text { for all }\ 0<t<T_1, \end{aligned}$$

with \(3<p\le \infty \) and \(c_0\) is a postive constant, then we show that \((\textbf{u},Q)\) preserves the energy equality on the closed interval \([0,T_{1}]\) including the first blow-up time \(T_{1}\).