From Random Advection to Transport Noise and the LU Framework for the Navier–Stokes Equations

We consider the incompressible Navier–Stokes equations in a smooth bounded domain in dimension two or three, with Dirichlet boundary conditions, when the velocity is advected by a rapidly varying stationary Markov process \eps^{-1}m(t/\eps^2). The fast field may be spatially inhomogeneous. We prove that, when \eps\to0, the laws of the large-scale velocities are tight and that their limit points are martingale solutions of the stochastic Navier–Stokes equations with transport noise in the form of the location uncertainty framework. The limiting equation contains in particular a noise which is described in terms of the covariance of the fast advection. Under a symmetry assumption on the temporal correlations of the fast field, the noise is Stratonovich and an advection by the Itô–Stokes drift appears. The proof relies on perturbed test functions, which we use both for the uniform energy bounds and for the identification of the limit. We have to adapt this method and add the Stokes operator to the Poisson equation associated to the fast advection in order to use fewer regularity assumptions and obtain correctors which satisfy the boundary conditions.