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Inverse problem for the geometric Navier-Stokes equations

Yavar Kian, Lauri Oksanen, Ziyao Zhao

TL;DR

The paper tackles the inverse problem of recovering a compact Riemannian manifold with boundary from local, fixed-time velocity observations of the Navier–Stokes flow driven by a local source. It reduces the nonlinear NS problem to a linearized Stokes setting and then to an auxiliary hyperbolic Stokes problem, enabling a Boundary Control (BC) method approach on a time-domain framework. By establishing local-to-global reconstruction via spectral data and conditional finite-speed of propagation, the authors show that the local source-to-solution data determine the manifold up to isometry. The result extends BC methods to vector-valued Stokes systems on manifolds with boundary and provides a pathway to recover the entire geometric structure from limited interior measurements. This has potential implications for geometric fluid-dynamics identification in geophysical, cosmological, and biological contexts where the ambient geometry is unknown. All mathematical notation is preserved with $...$ delimiters to aid precise interpretation and reuse in subsequent analyses.

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.

Inverse problem for the geometric Navier-Stokes equations

TL;DR

The paper tackles the inverse problem of recovering a compact Riemannian manifold with boundary from local, fixed-time velocity observations of the Navier–Stokes flow driven by a local source. It reduces the nonlinear NS problem to a linearized Stokes setting and then to an auxiliary hyperbolic Stokes problem, enabling a Boundary Control (BC) method approach on a time-domain framework. By establishing local-to-global reconstruction via spectral data and conditional finite-speed of propagation, the authors show that the local source-to-solution data determine the manifold up to isometry. The result extends BC methods to vector-valued Stokes systems on manifolds with boundary and provides a pathway to recover the entire geometric structure from limited interior measurements. This has potential implications for geometric fluid-dynamics identification in geophysical, cosmological, and biological contexts where the ambient geometry is unknown. All mathematical notation is preserved with delimiters to aid precise interpretation and reuse in subsequent analyses.

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.
Paper Structure (16 sections, 31 theorems, 274 equations)

This paper contains 16 sections, 31 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Comparision to previous literature
  4. Outline of the paper
  5. Preliminaries
  6. Stationary Stokes system
  7. Nonstationnary Navier-Stokes system and linearization
  8. Recovery of local spectral projections for the Stokes system
  9. Recovery of the local source-to-final value operator of the Stokes system
  10. Determination of the local spectral projections from the Stokes system
  11. Auxiliary hyperbolic Stokes problem
  12. Proof of Theorem \ref{['thm:main']}
  13. Reconstruction of metric in the accessible domain
  14. Determination of a relation among geodesic balls
  15. Reconstruction of metric in the domain of influence
  16. ...and 1 more sections

Key Result

theorem 1.1

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two n-dimensional smooth connected Riemannian manifolds with boundary. Let $\omega_1\subset M_1$ and $\omega_2\subset M_2$ be open and connected. For $j=1,2$, let $\mathcal{N}_j$ be the local source-to-final value map eq:def_N with $(M,g)=(M_j,g_j)$ and $\omega=\om Then $(M_1,g_1)$ and $(M_2,g_2)$ are isometric.

Theorems & Definitions (61)

  • theorem 1.1
  • remark 1.2
  • remark 2.1
  • lemma 2.2
  • proof
  • lemma 3.1
  • proof
  • lemma 3.2
  • lemma 3.3
  • proof
  • ...and 51 more