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

Thumbnail Image

Files

2211.15772.pdf (354.46 KB)

Links to Files

https://arxiv.org/abs/2211.15772

Permanent Link

https://doi.org/10.48550/arXiv.2211.15772
 http://hdl.handle.net/11603/30271

Collections

UMBC Mathematics and Statistics Department
UMBC Faculty Collection

Author/Creator

Author/Creator ORCID

https://orcid.org/0000-0001-8594-0568
 https://orcid.org/0000-0003-2764-4538

Date

2023-02-22

Type of Work

journal articles
preprints
Text

Department

Program

Citation of Original Publication

Rights

This item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.
Attribution 4.0 International (CC BY 4.0 DEED)

Subjects

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.