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

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 provide a 𝒑𝒓𝒊𝒐𝒓𝒊 verifiable conditions that ensure its convergence.