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Exact non-stationary solutions of the Euler equations in two and three dimensions

Patrick Heslin, Stephen C. Preston

TL;DR

This work develops an Arnold-geometric mechanism to construct explicit, smooth, global-in-time, typically non-stationary solutions to the incompressible Euler equations on 2D and 3D manifolds by exploiting a generalized Coriolis force spectrum. A key result shows that if a steady base flow $u_0$ and a complex perturbation $z$ satisfy simultaneous spectral relations with the inertia operator and coadjoint structure, then $U(t)=u_0+e^{i(\lambda-\zeta)t}z$ yields a real Euler flow and a related linearized flow, with nonstationarity captured by $\lambda\neq\zeta$. In 2D the authors obtain a complete curvature-based classification and explicit Kelvin and Rossby-Haurwitz-type solutions on flat, spherical, and hyperbolic domains, while in 3D they develop circle bundle and torus bundle geometries to produce new explicit nonstationary flows on manifolds such as $S^3$ via curl eigenfields and Chandrasekhar-Kendall-type constructions. The Euler-Arnold framework then clarifies how a generalized Coriolis force sits inside the geometric formulation and provides a concrete path to systematic nonstationary solutions, with clear avenues for stability analysis and extensions to related dynamics such as MHD and Navier-Stokes on manifolds.

Abstract

We develop, via Arnold's geometric framework, a mechanism for constructing explicit, smooth, global-in-time, and typically non-stationary solutions of the incompressible Euler equations. The approach introduces a notion of generalized Coriolis force, whose spectrum underlies the construction of these solutions. In the setting of ideal hydrodynamics, the construction recovers classical exact solutions such as Kelvin and Rossby-Haurwitz waves, while also producing new explicit examples on curved surfaces and three-dimensional manifolds including the round three-sphere. We further obtain a complete classification in two dimensions and a partial classification in three dimensions of the Riemannian manifolds that admit such solutions. The method is also formulated in the general Euler-Arnold setting and yields a simple criterion for non-stationarity.

Exact non-stationary solutions of the Euler equations in two and three dimensions

TL;DR

This work develops an Arnold-geometric mechanism to construct explicit, smooth, global-in-time, typically non-stationary solutions to the incompressible Euler equations on 2D and 3D manifolds by exploiting a generalized Coriolis force spectrum. A key result shows that if a steady base flow and a complex perturbation satisfy simultaneous spectral relations with the inertia operator and coadjoint structure, then yields a real Euler flow and a related linearized flow, with nonstationarity captured by . In 2D the authors obtain a complete curvature-based classification and explicit Kelvin and Rossby-Haurwitz-type solutions on flat, spherical, and hyperbolic domains, while in 3D they develop circle bundle and torus bundle geometries to produce new explicit nonstationary flows on manifolds such as via curl eigenfields and Chandrasekhar-Kendall-type constructions. The Euler-Arnold framework then clarifies how a generalized Coriolis force sits inside the geometric formulation and provides a concrete path to systematic nonstationary solutions, with clear avenues for stability analysis and extensions to related dynamics such as MHD and Navier-Stokes on manifolds.

Abstract

We develop, via Arnold's geometric framework, a mechanism for constructing explicit, smooth, global-in-time, and typically non-stationary solutions of the incompressible Euler equations. The approach introduces a notion of generalized Coriolis force, whose spectrum underlies the construction of these solutions. In the setting of ideal hydrodynamics, the construction recovers classical exact solutions such as Kelvin and Rossby-Haurwitz waves, while also producing new explicit examples on curved surfaces and three-dimensional manifolds including the round three-sphere. We further obtain a complete classification in two dimensions and a partial classification in three dimensions of the Riemannian manifolds that admit such solutions. The method is also formulated in the general Euler-Arnold setting and yields a simple criterion for non-stationarity.
Paper Structure (12 sections, 17 theorems, 200 equations)

This paper contains 12 sections, 17 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Ideal Hydrodynamics
  3. Two-Dimensional Fluids
  4. Three-Dimensional Fluids
  5. Circle Bundles over a Surface
  6. Geometric theory of circle bundles
  7. Curl eigenfields in a circle bundle
  8. Torus Bundles over an Interval
  9. The form of the metric
  10. Curl eigenfields in a torus bundle
  11. The Euler-Arnold Framework and Gerneralized Coriolis Force
  12. Future Directions

Key Result

Lemma 2.1

Let $(M,g)$ be a two- or three-dimensional compact Riemannian manifold, possibly with boundary. The inertia operator $\mathcal{A}$ given in inertia operator is invertible on $\mathfrak{X}_{\mu,\text{ex}}(M)$ and its inverse $\mathcal{A}^{-1}$ is compact and self-adjoint in $L^2$. Consequently, $\mat

Theorems & Definitions (49)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • proof : Proof of Theorem \ref{['simultaneous eigenvalue to euler solution']}
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Lemma 2.7
  • ...and 39 more