McKean-Vlasov type stochastic differential equations arising from the random vortex method

TL;DR

This paper establishes existence and uniqueness results for a class of McKean-Vlasov stochastic differential equations with singular kernels, relevant to fluid dynamics, using a novel approach that overcomes traditional Lipschitz continuity obstacles.

Contribution

The paper introduces a new method to prove existence and uniqueness for McKean-Vlasov SDEs with singular integral kernels, extending the theory to non-Lipschitz cases.

Findings

Proved existence and uniqueness of solutions for McKean-Vlasov SDEs with Biot-Savart kernel.

Developed a new analytical approach for distribution-dependent SDEs with singular coefficients.

Addressed challenges in applying classical SDE methods to fluid mechanics models.

Abstract

We study a class of McKean-Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and strong solutions for the McKean-Vlasov stochastic differential equations whose coefficients are defined in terms of singular integral kernels such as the Biot-Savart kernel. These SDEs which involve the distributions of solutions are in general not Lipschitz continuous with respect to the usual distances on the space of distributions such as the Wasserstein distance. Therefore there is an obstacle in adapting the ordinary SDE method for the study of this class of SDEs, and the conventional methods seem not appropriate for dealing with such distributional SDEs which appear in applications such as fluid mechanics.