A local-in-time theory for singular SDEs with applications to fluid models with transport noise

TL;DR

This paper develops a local-in-time theoretical framework for singular stochastic differential equations in Hilbert spaces, with applications to fluid models with transport noise, enabling existence, uniqueness, and blow-up criteria.

Contribution

It introduces a novel approximation approach for singular SDEs, applicable to fluid dynamics models with transport noise, including the stochastic CH and CCF systems.

Findings

Established local existence and uniqueness for singular SDEs.

Applied theory to stochastic fluid models with transport noise.

Derived blow-up criteria for specific fluid systems.

Abstract

In this paper, we establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in some Hilbert space. The key requirement is an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. With a cancellation estimate for generalized Lie derivative operators, we can construct such regular approximations for cases involving the Lie derivative operators, or more generally, differential operators of order one with suitable coefficients. In particular, we apply the abstract theory to achieve novel local-in-time results for the stochastic two-component Camassa--Holm (CH) system and for the stochastic C\'ordoba-C\'ordoba-Fontelos (CCF) model.