3D shear flows driven by Lévy noise at the boundary

Abstract

This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries. The fluid is driven by the noisy movement of the bottom boundary, where the noise is given by a Lévy process. After establishing existence of a martingale solution, we use the background flow method to derive an upper bound on the turbulent energy dissipation rate. Our estimate recovers one of the basic scaling ideas of turbulence theory, namely, that the dissipation rate is independent of the viscosity at high Reynolds number.