论文标题
关于从2D Navier-Stokes方程中低模式观测的未知驱动力重建
On the reconstruction of unknown driving forces from low-mode observations in the 2D Navier-Stokes Equations
论文作者
论文摘要
本文涉及确定来自流动磁场上的大规模观察的不可压缩流体的非电位，外部时间依赖性扰动的未知来源的问题。提出了一种基于松弛的方法来实现这一目标，该方法利用运动方程的非线性特性将渐近的小尺度奴役到大尺度上。特别是，引入了一种算法，该算法系统地在未观察到的尺度上系统地产生流场的近似值，以便生成与未知力的近似值。然后重复该过程以生成改进的未观察到的尺度的近似值，依此类推。该算法收敛的数学证明是在具有周期性边界条件的二维Navier-Stokes方程的背景下建立的，假设该力属于相位空间的观察性子空间；在算法中的每个阶段，都表明该模型误差表示为近似和真实力之间的差异，以几何方式渐近地减少至零，只要观察到足够多的量表，并且适当调节了算法的某些参数；还讨论了假设的清晰度问题，包括更新之间的瞬时时期等其他考虑因素。
This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which leverages a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier-Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decrements to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned; the issue of the sharpness of the assumptions, among other practical considerations such as the transient periods between updates, are also discussed.
