带局域噪声的随机Navier-Stokes方程的指数混合
Exponential mixing for the stochastic Navier--Stokes equation with localized noise
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中文总结 AI 辅助
本文针对带局域噪声的二维Navier-Stokes方程，结合Malliavin微积分与PDE控制理论，证明了相关Navier-Stokes-热系统的指数混合及速度律的指数收敛，为解决噪声退化问题提出了稳定转换方案。
中文摘要 AI 辅助
本文研究有界区域上满足Navier滑移边界条件的二维Navier-Stokes方程，该方程由随机热方程产生的空间局域平稳外力驱动。我们证明了相关Navier-Stokes-热系统的指数混合，进而得到平稳外力下速度律的指数收敛。主要难点在于白噪声被限制在子区域内，且通过热分量间接作用于速度。为解决退化问题，证明结合了Malliavin微积分与PDE控制理论，关键要素是一种稳定方案，可将局域Navier-Stokes控制转换为与热动力学兼容的时间正则控制。
英文摘要
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.
发表机构
- University of Science and Technology Beijing（北京科技大学）
- Peking University（北京大学）
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