退化布朗强迫下随机原始方程的谱间隙
Spectral gap for the stochastic primitive equations under degenerate Brownian forcing
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中文总结 AI 辅助
针对退化布朗强迫下的三维原始方程，提出定量稳定-紧渐近耦合准则，证明加权Wasserstein谱间隙、唯一遍历性与指数混合，并给出最优强迫方向数。
中文摘要 AI 辅助
我们在$\mathscr E=\{u\in H^1\cap L^8: \partial_zu\in L^4\}$上证明了三维原始方程在有限秩加性布朗强迫下的加权Wasserstein谱间隙，该强迫是相位完备的有限傅里叶型，并满足定量二次饱和条件。特别地，四个实强迫方向就足够了，且该数目在相位完备类中是最优的。因此，该系统在$H^1$拓扑下是唯一遍历且指数混合的。主要困难源于缺乏足够强的矩界以及噪声的高度退化性，这使得问题超出了标准方法的直接适用范围。我们引入了一种新的定量稳定-紧渐近耦合准则，该准则捕捉了初始扰动增长的可用控制与通过扰动噪声来补偿它们的精度-成本之间的微妙平衡。对于原始方程，这导致了一种定量的无穷维Hörmander分析，需要跟踪具有傅里叶频率的Lie括号多项式的阶数和系数成本。
英文摘要
We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on $\mathscr E=\{u\in H^1\cap L^8: \partial_zu\in L^4\}$ under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forcing directions suffice and this number is optimal within the phase-complete class. Consequently, the system is uniquely ergodic and exponentially mixing in the $H^1$ topology. The main difficulties stem from the lack of sufficiently strong moment bounds and the highly degenerate nature of the noise, placing the problem outside the direct reach of standard methods. We introduce a new quantitative stable--compact asymptotic coupling criterion that captures the delicate balance between the available control of the growth of initial perturbations and the accuracy--cost of compensating them by perturbing the noise. For the primitive equations, this leads to a quantitative infinite-dimensional Hörmander analysis requiring tracking the degree and coefficient cost of Lie-bracket polynomials with the Fourier frequency.
发表机构
- Clemson University（克莱姆森大学）
- Sichuan University（四川大学）
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