Partial Regularity for the Three-dimensional Stochastic Ericksen--Leslie equations
Hengrong Du, Chuntian Wang
TL;DR
The paper addresses the global existence and partial regularity of 3D stochastic Ericksen--Leslie equations on the torus with additive noise by introducing martingale suitable weak solutions. It employs a split-up scheme, decomposing the velocity into a linear stochastic Stokes part and a nonlinear modified Ericksen--Leslie part, and constructs an energy-preserving approximation to pass to the limit via Skorokhod embedding. A blow-up analysis, aided by an almost boundedness result and the A-B-C-D criteria in Morrey-type spaces, yields a parabolic-1D Hausdorff measure zero result for the singular set. The findings extend partial regularity theory to stochastic complex fluids and provide a robust framework potentially applicable to other stochastically forced PDEs in fluid dynamics.
Abstract
In this article, we investigate the global existence of martingale suitable weak solutions to stochastic Ericksen-Leslie equations with additive noise in a 3D torus. The notion of suitable weak solutions has been introduced to address possible emergence of finite-time singularities, which remains a notably challenging question in the field of fluid dynamics. Weak solutions offer an approach to account for these potential singularities. A restricted class of weak solutions that exhibit a higher level of regularity, and are therefore more likely to be physically meaningful, is naturally called for. Consequently, suitable weak solutions -- that is, weak solutions that satisfy a local energy inequality -- have become a focus of research, including investigations into how regular these solutions can be. In this article, we prove that, despite the presence of white noise, the paths of martingale suitable weak solutions of 3D stochastic Ericksen-Leslie equations exhibit singular points of one-dimensional parabolic Hausdorff measure zero. To establish this result, we utilize two techniques that can potentially be generalized to handle other stochastically forced complex fluid dynamics equations with a similar structure. First, a local energy-preserving approximation is constructed, which markedly facilitates the proof of the global existence of martingale suitable weak solutions. Second, to demonstrate partial regularity of these solutions, a blow-up argument is formulated, which efficiently yields the desired key estimate.