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Thin shell limit and the derivation of the viscosity operator on the ellipsoid

Chi Hin Chan, Magdalena Czubak, Padi Fuster Aguilera

TL;DR

This work analyzes the thin shell limit around an ellipsoid to derive intrinsic viscosity operators governing velocity fields on the surface. By performing asymptotic expansions in the scaling and normal directions and imposing Navier-slip or homogeneous Hodge boundary conditions, the authors obtain operator forms that intertwine Bochner/Laplacian-type terms with Lie-derivative and curvature contributions. A key finding is that the resulting intrinsic operator on the ellipsoid can be expressed either as a deformation-Laplacian-like term with a Lie-bracket correction, or as a Hodge-Laplacian with an accompanying Lie derivative, depending on the boundary condition and expansion direction. The work also provides geometric interpretations of Navier and Hodge conditions via Lie derivatives and recovers known results in the spherical limit, highlighting how curvature and averaging direction affect the viscosity operator on curved manifolds.

Abstract

In this paper we derive four new candidates for an intrinsic viscosity operator on an ellipsoid by using the heuristic of the thin shell limit along the scaling direction of the ellipsoid. We show that the general method of the thin shell limit through the asymptotic expansion depends on the averaging method used. We consider both the homogeneous Navier and Hodge boundary conditions. We also obtain a geometric representation of these two boundary conditions.

Thin shell limit and the derivation of the viscosity operator on the ellipsoid

TL;DR

This work analyzes the thin shell limit around an ellipsoid to derive intrinsic viscosity operators governing velocity fields on the surface. By performing asymptotic expansions in the scaling and normal directions and imposing Navier-slip or homogeneous Hodge boundary conditions, the authors obtain operator forms that intertwine Bochner/Laplacian-type terms with Lie-derivative and curvature contributions. A key finding is that the resulting intrinsic operator on the ellipsoid can be expressed either as a deformation-Laplacian-like term with a Lie-bracket correction, or as a Hodge-Laplacian with an accompanying Lie derivative, depending on the boundary condition and expansion direction. The work also provides geometric interpretations of Navier and Hodge conditions via Lie derivatives and recovers known results in the spherical limit, highlighting how curvature and averaging direction affect the viscosity operator on curved manifolds.

Abstract

In this paper we derive four new candidates for an intrinsic viscosity operator on an ellipsoid by using the heuristic of the thin shell limit along the scaling direction of the ellipsoid. We show that the general method of the thin shell limit through the asymptotic expansion depends on the averaging method used. We consider both the homogeneous Navier and Hodge boundary conditions. We also obtain a geometric representation of these two boundary conditions.

Paper Structure

This paper contains 29 sections, 12 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Coordinates on the ellipsoid and on $\mathbb R^3$: scaling direction
  5. Shape operator
  6. Divergence free conditions
  7. Coordinates on the ellipsoid and on $\mathbb R^3$: normal direction
  8. Helpful formulas
  9. Power series expansion
  10. Proof of Theorem \ref{['thmb']}: geometric formulation of boundary conditions
  11. Navier condition as a Lie derivative
  12. Hodge condition as a Lie derivative
  13. Proof of Theorems \ref{['thm1']} and \ref{['thmd']}: expansion in the scaling direction
  14. Viscosity operator: deriving an intrinsic formula
  15. Bochner Laplacian and the Ricci operator
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $a,\varepsilon>0$, and $E=\{{x^2+y^2+a^2z^2}=a^2\}$ be the ellipsoid embedded in $\mathbb R^3$, and $E_{1+\varepsilon}=\{{x^2+y^2+a^2z^2}=a^2(1+\varepsilon)\}$ the rescaled ellipsoid. Given a vector field $v$ in the thin shell with boundary $E \cup E_{1+\varepsilon}$, where $v$ is also assumed t for the perfect Navier-slip boundary condition, and for the homogeneous Hodge boundary condition,

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 18 more