Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations

TL;DR

This paper introduces a new class of distribution-flow dependent SDEs motivated by Navier-Stokes equations, establishing their well-posedness and exploring their connection to PDEs and fractional Brownian motion.

Contribution

It proposes a novel distribution-flow dependent SDE framework, proves existence and uniqueness under broad conditions, and analyzes a stochastic 2D-Navier-Stokes equation with fractional noise.

Findings

Existence and uniqueness of solutions under one-sided Lipschitz conditions.

Global well-posedness for 2D-Navier-Stokes with fractional Brownian noise for Hurst parameter in (0, 1/2).

Solutions are smooth when initial vorticity is a finite signed measure.

Abstract

Motivated by the probabilistic representation for solutions of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on the entire flow of its time marginals. We establish the existence and uniqueness of both strong and weak solutions under one-sided Lipschitz conditions and for singular drifts. These newly proposed distribution-flow dependent stochastic differential equations are closely connected to quasilinear backward Kolmogorov equations and Fokker-Planck equations. Furthermore, we investigate a stochastic version of the 2D-Navier-Stokes equation associated with fractional Brownian noise. We demonstrate the global well-posedness and smoothness of solutions when the Hurst parameter $H$ lies in the range $(0, \frac12)$ and the initial vorticity is a finite signed measure.