Inverse problem for the geometric Navier-Stokes equations

TL;DR

This paper addresses the inverse problem of identifying a Riemannian manifold with boundary from localized observations of Navier-Stokes solutions, using a novel reduction to a hyperbolic Stokes system and an extended Boundary Control method.

Contribution

It introduces a new approach that reduces the inverse problem for Navier-Stokes equations to a hyperbolic Stokes system and generalizes the Boundary Control method for this context.

Findings

Successfully reduces the inverse problem to a hyperbolic Stokes system.

Develops a generalized Boundary Control method for the problem.

Provides a framework for determining manifold geometry from localized fluid flow data.

Abstract

We consider the inverse problem of determining a compact Riemannian manifold with boundary from fixed time observations of the solution, restricted to a small subset in space, for the Navier-Stokes system with a local source on the manifold. Our approach is based on a reduction to an inverse problem for an auxiliary hyperbolic Stokes system, via linearization and spectral techniques. We solve the resulting inverse problem by a new generalization of the Boundary Control method.