Regularity for a Fractional Liquid Crystal Model with Anomalous Dissipation and Thermal Effects

In this paper, we present proofs of the regularity criterion for weak solutions to a generalized liquid crystal model with fractional diffusion and thermal effects. Specifically, we prove that if certain decay estimates hold, the solutions \((u, d, \theta )\), representing the velocity field, the orientation vector in the crystal and the temperature field respectively, are regular at the origin. We employ the extension technique for the fractional Laplacian and establish decay estimates to demonstrate the boundedness and regularity of the solutions. Our results aim to extend classical theories by introducing anomalous dissipation and thermal influences, leading to potentially increase the framework for the analysis of liquid crystal models under varying conditions.