Convergence Analysis of a Viscosity Parameter Recovery Algorithm for the 2D Navier-Stokes Equations

TL;DR

This paper proves the convergence of a feedback control algorithm for estimating the viscosity in 2D Navier-Stokes equations, using continuous observations and sensitivity analysis, marking a novel result for nonlinear PDE parameter estimation.

Contribution

It provides the first convergence proof for a viscosity recovery algorithm in nonlinear PDEs using continuous-time feedback control and sensitivity bounds.

Findings

Convergence achieved under a natural non-degeneracy condition.

Analysis covers two parameter update rules: instantaneous and time-averaged.

Dissipative structure identified for the sensitivity variable's time-derivative.

Abstract

In this paper, the convergence of an algorithm for recovering the unknown kinematic viscosity of a two-dimensional incompressible, viscous fluid is studied. The algorithm of interest is a recursive feedback control-based algorithm that leverages observations that are received continuously-in-time, then dynamically provides updated values of the viscosity at judicious moments. It is shown that in an idealized setup, convergence to the true value of the viscosity can indeed be achieved under a natural and practically verifiable non-degeneracy condition. This appears to be first such result of its kind for parameter estimation of nonlinear partial differential equations. Analysis for two parameter update rules is carried out: one which involves instantaneous evaluation in time and the other, averaging in time. The proofs of convergence for either rule exploits sensitivity-type bounds in higher-order Sobolev topologies, while the instantaneous version particularly requires delicate energy estimates involving the time-derivative of the sensitivity-type variable. Indeed, a crucial component in the analysis of the first update rule is the identification of a dissipative structure for the time-derivative of the sensitivity-type variable, which ultimately ensures a favorable dependence on the tuning parameter of the algorithm.