The Fractal Dimension of Singularities for Liquid Crystals Inequalities Without Maximum Principle

In this paper, we investigate partial regularity results for the so-called solutions to the “liquid crystals inequalities". These solutions essentially satisfy the incompressible condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the momentum conservation equations, but are not required to satisfy the simplified Ericksen–Leslie system (for the liquid crystal flows) itself. We prove that under an additional assumption on the orientation \(\textbf{d}\) $\mathbf{d}$ for any fixed value of a certain parameter \(q\in (5,6)\) $q\in (5,6)$ , the solutions to the corresponding “liquid crystals inequalities", which may lack the maximum principle for \(\textbf{d}\) $\mathbf{d}$ , satisfy \(\dim _{f}(\mathcal {K}\cap \Sigma )\le \frac{360}{277}(\thickapprox 1.2996)\) ${dim}_f(\mathcal{K}\cap \mathrm{\Sigma })\le \frac{360}{277}(\approx 1.2996)$ . Here, \(\dim _{f}\) ${dim}_f$ is the parabolic fractal dimension, \(\mathcal {K}\subseteq \Omega _T\) $\mathcal{K}\subseteq {\mathrm{\Omega }}_T$ is any compact set, and \(\Sigma \) $\mathrm{\Sigma }$ is the set of forward-sigular space-time points near which the solution blows-up forwards in time.