Partial Regularity for the Three-dimensional Stochastic Ericksen--Leslie equations

Abstract: In this article, we investigate the existence of martingale suitable weak solutions to 3D stochastic Ericksen--Leslie equations driven by an external random force. Despite the inherent white noise, we prove that paths of martingale suitable weak solutions exhibit singular points of one-dimensional parabolic Hausdorff measure zero. Our approach is mainly based on two ideas: a local energy-preserving approximation which serves to establish the existence of suitable weak solutions, and a blow-up argument employed to achieve the partial regularity.