Determining the viscosity of the Navier-Stokes equations from observations of finitely many modes

TL;DR

This paper investigates the conditions under which the viscosity of a fluid governed by Navier-Stokes equations can be uniquely identified from limited velocity measurements, providing theoretical guarantees and an algorithm for the inverse problem.

Contribution

It introduces a mathematical framework for the inverse problem of determining viscosity from sparse data, with explicit conditions for well-posedness and an algorithm with proven convergence.

Findings

Explicit a priori conditions for well-posedness.

Small loss functional implies closeness to true viscosity.

Algorithm converges to the true viscosity under certain conditions.

Abstract

In this work, we ask and answer the question: when is the viscosity of a fluid uniquely determined from spatially sparse measurements of its velocity field? We pose the question mathematically as an optimization problem using the determining map (the mapping of data to an approximation made via a nudging algorithm) to define a loss functional, the minimization of which solves the inverse problem of identifying the true viscosity given the measurement data. We give explicit a priori conditions for the well-posedness of this inverse problem. In addition, we show that smallness of the loss functional implies proximity to the true viscosity. We then present an algorithm for solving the inverse problem and prove its convergence.