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 H1 data, together with smallness of the space-time L6-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 L6t,x.
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BibTeX
@misc{brandt2026regularizing,
title = {{Regularizing the Navier-Stokes Equations by One Noise Component}},
author = {Felix Brandt and Igor Kukavica},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02557},
eprint = {2611.02557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02557},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Regularizing the Navier-Stokes Equations by One Noise Component
AU - Felix Brandt
AU - Igor Kukavica
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02557
UR - https://arxiv.org/abs/2611.02557
AB - 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 H^1 data, together with smallness of the space-time 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 L^6_t,x.
ER -