Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field

TL;DR

This paper presents a new continuous data assimilation algorithm for the 2D Navier-Stokes equations that uses measurements of only one velocity component, ensuring exponential convergence to the true solution under certain resolution conditions.

Contribution

It introduces a novel assimilation method that requires only partial velocity data and provides rigorous conditions for convergence to the true flow.

Findings

Algorithm achieves exponential convergence to the true solution.

Applicable with various types of finite-dimensional observables.

Conditions on data resolution ensure accuracy and stability.

Abstract

We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier-Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier-Stokes equations, associated with the observed (finite dimensional projection of) velocity.