Regularizing the Navier-Stokes Equations by One Noise Component

Felix Brandt; Igor Kukavica

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arXiv · 2611.02557

Regularizing the Navier-Stokes Equations by One Noise Component

Felix Brandt·Igor Kukavica

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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 $\mathrm{L}^6_{t,x}$.

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Felix Brandt, Igor Kukavica. 2026-11-02. Regularizing the Navier-Stokes Equations by One Noise Component. https://arxiv.org/abs/2611.02557

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