发表机构
Dalian University of Technology（大连理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对环面上带快速振荡系数的分布依赖随机Navier--Stokes方程，建立了三个弱平均原理，证明了弱拉回吸引子上半连续收敛到平均方程的弱全局吸引子。
AI 中文摘要
我们在环面$\T^2$上建立了具有快速振荡系数的分布依赖随机Navier--Stokes方程的三个弱平均原理。首先，在有限时间区间上，解定律是相对紧的，并且当$\var\to0$时，每个极限点都是平均方程的变分鞅解的路径定律。在耗散条件下，我们可以选择原始方程和平均方程的有界完备变分解定律，使得每个满足$\var\to0$的序列都有一个子序列弱收敛到平均方程的有界完备变分解定律。在概率定律层面上，原始非自治方程拥有一族弱拉回吸引子，而平均方程则有一个弱全局吸引子。弱拉回吸引子上半连续地收敛到平均方程的弱全局吸引子，且关于系数壳一致收敛。
英文摘要
We establish three weak averaging principles for distribution-dependent stochastic Navier--Stokes equations with rapidly oscillating coefficients on the torus $\T^2$. First, solution laws are relatively compact on finite time intervals, and every limit point as $\var\to0$ is the path law of a variational martingale solution of the averaged equation. Under a dissipative condition, we can choose bounded complete variational solution laws of the original and averaged equations so that every sequence with $\var\to0$ has a subsequence converging weakly to a bounded complete variational solution law of the averaged equation. At the level of probability laws, the original nonautonomous equation admits a family of weak pullback attractors, while the averaged equation has a weak global attractor. The weak pullback attractors converge upper semicontinuously to the weak global attractor of the averaged equation, uniformly with respect to the coefficient hull.
