带有非退化噪声的二维Navier–Stokes方程投影过程的唯一遍历性
Unique Ergodicity for the Projective Process of the 2D Navier--Stokes Equation with Nondegenerate Noise
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中文总结 AI 辅助
该研究证明了带非退化加性对角噪声的二维涡度形式Navier–Stokes方程投影过程的唯一遍历性，推导了对应的Furstenberg–Khasminskii公式，提出了无Foiaş–Prodi分解时的渐近广义耦合机制，构造了Wiener路径的有限秩扰动实现局部指数收缩。
中文摘要 AI 辅助
我们证明了二维涡度形式的Navier–Stokes方程所关联的投影过程的唯一遍历性，该方程带有作用于每个非零实傅里叶相位的加性对角噪声，且满足双侧幂律界。由此，关于最大Lyapunov指数的精确Furstenberg–Khasminskii公式成立。主要新要素是渐近广义耦合的紧致稠密机制，该机制适用于投影动力学不存在Foiaş–Prodi型高低模态分解的情形。利用Malliavin导数的稠密性和状态导数的紧致性，我们构造了Wiener路径的有限秩扰动，以一阶补偿初始条件的扰动，留下的残差其对数增长具有负的平稳均值，因此以指数速率局部收缩，其代价由分块Ramer变换的三角方案控制。
英文摘要
We prove unique ergodicity of the projective process associated with the two dimensional Navier--Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg--Khasminskii formula for the top Lyapunov exponent holds. The main new ingredient is a compact dense mechanism for asymptotic generalized coupling in the absence of a Foiaş--Prodi type high-low mode decomposition for the projective dynamics. Using the dense range of the Malliavin derivative and compactness of the state derivative, we construct finite rank perturbations of the Wiener path that compensate, to first order, for perturbations of the initial condition, leaving a residual whose logarithmic growth has negative stationary mean and hence contracts locally at an exponential rate, with a cost controlled by a triangular scheme of blockwise Ramer transformations.
