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

Motivated by probabilistic representations of the Navier–Stokes equations, we introduce a new class of stochastic differential equations whose coefficients depend on the entire flow of time marginals. Under one-sided Lipschitz conditions and in the presence of singular drifts, we establish the existence and uniqueness of both strong and weak solutions. These distribution-flow dependent stochastic differential equations are closely related to quasilinear backward Kolmogorov equations and Fokker–Planck equations. We also study distribution-flow dependent SDEs associated with generalized two-dimensional Navier–Stokes equations and driven by fractional Brownian noise. When the Hurst parameter H ∈ (0,¹2^{)and the initial vorticityν0∈ M(R2}), we prove global well-posedness and smoothness of solutions. In the case H =¹2^{, we} establish global strong well-posedness and smoothness for the corresponding SDEs associated with the backward Navier–Stokes equations, and clarify their relation to the Navier–Stokes equations.