Regularizing the Navier-Stokes Equations by One Noise Component

Abstract

We study the three-dimensional incompressible Navier--Stokes equations on the torus with multiplicative Itô noise which is driven by a single Brownian motion and acts only through the vertical velocity component. Such a noise is strongly degenerate: An infinite-dimensional part of the velocity field is not forced at all. We show that it nevertheless has a regularizing effect. For a linear noise, strong solutions whose initial data have no unforced part are global with probability tending to one as the noise intensity grows. For a suitable nonlinear noise, the same holds for arbitrary $\mathrm{H}^1$ data, together with smallness of the space-time $\mathrm{L}^6$-norm of the vertical velocity. As a deterministic counterpart, we give a self-contained proof that a quintic damping of the vertical component yields global regularity, and the same argument shows that, under the linear noise, blow-up can only occur if the vertical velocity leaves $\mathrm{L}^6_{t,x}$.