Fluctuation dynamics in randomly advected Navier-Stokes equations below critical scaling

2026/07/17 by Arnaud Debussche, Martina Hofmanová
#math.AP #math.PR

paper · pdf

Abstract

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale ε² and has spatial correlation length δ; the critical regime ε = δ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime ε = o (δ), we prove a law of large numbers in dimensions d = 2, 3: the solutions converge to a deterministic Navier--Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption ε = o (δ^{1 + ι}) for some ι> 0, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.

Related

• Emergent dynamical scaling in the inviscid limit of 3D stochastic Navier-Stokes equation with thermal noise
• Convergence of the stochastic Navier-Stokes-α solutions toward the stochastic Navier-Stokes solutions
• Diffusive hydrodynamics from long-range correlations
• Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations
• Hydrodynamic noise in one dimension: projected Kubo formula and how it vanishes in integrable models