Viscosity estimation for 2D pipe flows I. Construction, consistency, asymptotic normality

TL;DR

This paper develops a statistically consistent and asymptotically normal estimator for the viscosity of a 2D viscous fluid flow, based solely on enstrophy observations, under stochastic forcing conditions.

Contribution

It introduces a novel estimator for fluid viscosity in 2D flows and proves its strong consistency and asymptotic normality using explicit formulas and flow mixing properties.

Findings

Estimator is strongly consistent.

Estimator is asymptotically normal.

Applicable to flows with stochastic forcing.

Abstract

We consider the motion of incompressible viscous fluid in a rectangle, imposing the periodicity condition in one direction and the no-slip boundary condition in the other. Assuming that the flow is subject to an external random force, white in time and regular in space, we construct an estimator for the viscosity using only observations of the enstrophy. The goal of the paper is to prove that the estimator is strongly consistent and asymptotically normal. The proof of consistency is based on the explicit formula for the estimator and some bounds for trajectories, while that of asymptotic normality uses in addition mixing properties of the Navier-Stokes flow.