Stochastic Navier–Stokes Equations with Non-Markovian Noise
ABSTRACT
Stochasticity is an intrinsic feature of fluid dynamics. We study Navier–Stokes equations with random input from fast oscillatory random transport vector fields representing unresolved turbulent fluctuations. Although the underlying dynamics are differentiable in time, an effective limit produces multiplicative transport noise driven by fractional Brownian motion, with trajectories of Holder regularity below 1/2. This creates temporal roughness at the level of the effective equation, requiring a rough path interpretation with second-order correction terms. The effective limit thus exhibits a form of roughness creation: smooth-in-time random dynamics generate a genuinely rough stochastic transport equation. We report recent progress in this direction.
