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Exponential mixing for the stochastic Navier--Stokes equation with localized noise
Abstract
This paper studies the 2D Navier--Stokes equation on a bounded domain with Navier-slip boundary conditions, driven by spatially localized stationary forcing generated by a stochastic heat equation. We prove exponential mixing for the associated Navier--Stokes--heat system, and consequently exponential convergence of the velocity law under stationary forcing. The main difficulty is that the white noise is confined to a subdomain and reaches the velocity indirectly through the heat component. To address the degeneracy, the proof combines Malliavin calculus with PDE control theory. The key ingredient is a stabilization scheme that converts localized Navier--Stokes controls into time-regular controls compatible with the heat dynamics.
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BibTeX
@misc{liu2026exponential,
title = {{Exponential mixing for the stochastic Navier--Stokes equation with localized noise}},
author = {Ziyu Liu and Shengquan Xiang and Zhifei Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28964},
eprint = {2608.28964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28964},
}
RIS
TY - UNPB
TI - Exponential mixing for the stochastic Navier--Stokes equation with localized noise
AU - Ziyu Liu
AU - Shengquan Xiang
AU - Zhifei Zhang
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.28964
UR - https://arxiv.org/abs/2608.28964
AB - This paper studies the 2D Navier--Stokes equation on a bounded domain with Navier-slip boundary conditions, driven by spatially localized stationary forcing generated by a stochastic heat equation. We prove exponential mixing for the associated Navier--Stokes--heat system, and consequently exponential convergence of the velocity law under stationary forcing. The main difficulty is that the white noise is confined to a subdomain and reaches the velocity indirectly through the heat component. To address the degeneracy, the proof combines Malliavin calculus with PDE control theory. The key ingredient is a stabilization scheme that converts localized Navier--Stokes controls into time-regular controls compatible with the heat dynamics.
ER -