Continuous data assimilation for the three-dimensional Navier-Stokes-$\alpha$

TL;DR

This paper introduces a continuous data assimilation algorithm for the 3D Navier-Stokes-$ extalpha$ model, using observational data to synchronize large-scale solutions and ensure convergence to the true solution over time.

Contribution

It develops a nudging-based data assimilation method for the 3D Navier-Stokes-$ extalpha$ model with conditions guaranteeing convergence from observational data.

Findings

Conditions on data resolution ensure convergence of approximate to true solution.

The algorithm synchronizes large spatial scales with observational measurements.

Provides bounds based on physical parameters like viscosity and domain size.

Abstract

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present a continuous data assimilation algorithm for the three-dimensional Navier-Stokes-$\alpha$ model. This algorithm consists of introducing a nudging process through general type of approximation interpolation operator (that is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of the unknown solutions the Navier-Stokes-$\alpha$ equations that corresponds to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, that is obtained from these collected data, converges to the unkown reference solution (physical reality) over time. These conditions are given in terms of some physical parameters, such as kinematic viscosity, the size of the domain and the forcing term.