标题： 无网格插值可观测量用于连续数据同化 显示英文标题

标题： Mesh-Free Interpolant Observables for Continuous Data Assimilation

摘要： 本文考虑了一种基于校正的数据同化方案，用于具有周期性边界条件的二维（2D）纳维-斯托克斯方程（NSE），并研究了该算法产生的信号与对应观测数据的真实信号在所有高阶Sobolev拓扑中的同步性。 这项工作补充了文献中之前的结果，其中确定了在何种条件下可以保证同步性，要么仅针对$H^1$拓扑，当观测是通用观测时；要么针对解析Gevrey拓扑，当观测是谱观测时。 为了适应更强拓扑中的同步性特性，最初由Azouani、Olson和Titi引入的广义插值观测算子框架被扩展到一个更丰富的算子类。 大量努力被投入到这个更扩展的框架的发展中，具体包括：其基本逼近性质的识别、相关于获得同步性的这类算子的子类的识别，以及这些算子结构与系统关于同步性特性的详细关系。 该框架的主要特点之一是其“无网格”特性，这使得观测数据本身能够决定域的划分。 最后，得到了二维NSE在所有高阶Sobolev范数下的吸收球半径的估计，从而正确地推广了之前已知的界限；这样的估计对于在高阶拓扑中建立算法的同步性特性是必需的。 显示英文摘要

摘要： This paper considers a nudging-based scheme for data assimilation for the two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the synchronization of the signal produced by this algorithm with the true signal, to which the observations correspond, in all higher-order Sobolev topologies. This work complements previous results in the literature where conditions were identified under which synchronization is guaranteed either with respect to only the $H^1$--topology, in the case of general observables, or to the analytic Gevrey topology, in the case of spectral observables. To accommodate the property of synchronization in the stronger topologies, the framework of general interpolant observable operators, originally introduced by Azouani, Olson, and Titi, is expanded to a far richer class of operators. A significant effort is dedicated to the development of this more expanded framework, specifically, their basic approximation properties, the identification of subclasses of such operators relevant to obtaining synchronization, as well as the detailed relation between the structure of these operators and the system regarding the syncrhonization property. One of the main features of this framework is its "mesh-free" aspect, which allows the observational data itself to dictate the subdivision of the domain. Lastly, estimates for the radius of the absorbing ball of the 2D NSE in all higher-order Sobolev norms are obtained, thus properly generalizing previously known bounds; such estimates are required for establishing the synchronization property of the algorithm in the higher-order topologies.