Partial regularity and the upper Minkowski dimension of singularities for suitable weak solutions to the 3D co-rotational Beris-Edwards system

Abstract

We study partial regularity and the upper Minkowski dimension of potential singularities for suitable weak solutions to the 3d co-rotational Beris-Edwards system for the nematic liquid crystal flows with Landau-de Gennes potential. Precisely, we establish that there exists a $\epsilon >0$ such that if $(u,Q,P)$ is a suitable weak solution, and satisfies$r^{-\frac{6\alpha }{7\alpha -6}}{\int }_{t_0-r^2}^{t_0}{(\parallel (\left|u\right|}^2{,\left|\nabla Q\right|}^2{)\parallel }_{L^{\alpha }\left(B_r\left(x_0\right)\right)}^{\beta }+{\parallel P\parallel }_{L^{\alpha }\left(B_r\left(x_0\right)\right)}^{\beta })dt\le \epsilon ,$where $\alpha \in [\frac{6}{5},2]$ and $\beta =\frac{4\alpha }{7\alpha -6}\in [1,2]$, then $(u,Q)$ is regular at $z_0$. Based upon the regularity result above, we then prove the upper Minkowski dimension of the potential singularities for any suitable weak solution is at most $\frac{975}{758}(\approx 1.286)$. Additionally, if $(u,\nabla Q)\in L^p(0,T;L^q\left(R^3\right))$ with $1\le \frac{2}{p}+\frac{3}{q}$ and $\frac{20}{7}\le p,q<\infty$, then the upper Minkowski dimension of the potential singularities is no greater than $\max \{p,q\}(\frac{2}{p}+\frac{3}{q}-1)$.