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Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves

Giuseppe La Scala

TL;DR

The work addresses the 2D capillary drop with a free boundary by casting the dynamics as a Craig-Sulem-type system on $\mathbb{S}^1$ and, after a conformal torus reinterpretation, on $\mathbb{T}^1$. It unveils a Hamiltonian structure and establishes conserved quantities from torus and reversibility symmetries, enabling a rigorous bifurcation analysis. Rotating-wave solutions are shown to exist near the static circle via a Lyapunov–Schmidt reduction from a multiple-eigenvalue, with branches labeled by angular momentum and unique on each orbit; symmetric variants under reversibility and $c$-fold symmetries are also constructed. The results extend the capillary-drop framework to a 2D torus setting, providing a rigorous foundation for rotating drops and their symmetry-related multiplicity, with potential implications for understanding capillary-driven dynamics on curved manifolds. The analysis relies on a detailed treatment of the Dirichlet-Neumann operator and conformal geometry, yielding precise variational and Hamiltonian characterizations of the rotating waves.

Abstract

We consider the free boundary problem for a two-dimensional, incompressible, perfect, irrotational liquid drop of nearly circular shape with capillarity: that is, we consider the 2D version of the 3D capillary drop problem treated in Baldi-Julin-La Manna [11] and Baldi-La Manna-La Scala [12]. In particular, we derive its Craig-Sulem formulation firstly over the circle, then over the one-dimensional flat torus; the arising equations are similar to the pure capillary Water Waves for the ocean problem, apart from conformal factors and additional terms due to curvature terms. Then, we show its Hamiltonian structure and we derive constants of motions from symmetries, one of which is the invariance by the torus action. Thanks to this invariance, we show the existence of orbits of rotating wave solutions (which are the analogous of travelling waves of the ocean problem) by bifurcation from multiple eigenvalues in the spirit of Moser-Weinstein [44, 56] and Craig-Nicholls [22] variational approaches; in particular, we can parametrize such orbits by the angular momentum, and for each value of it they are unique. This will imply that each orbit is generated by symmetric rotating waves.

Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves

TL;DR

The work addresses the 2D capillary drop with a free boundary by casting the dynamics as a Craig-Sulem-type system on and, after a conformal torus reinterpretation, on . It unveils a Hamiltonian structure and establishes conserved quantities from torus and reversibility symmetries, enabling a rigorous bifurcation analysis. Rotating-wave solutions are shown to exist near the static circle via a Lyapunov–Schmidt reduction from a multiple-eigenvalue, with branches labeled by angular momentum and unique on each orbit; symmetric variants under reversibility and -fold symmetries are also constructed. The results extend the capillary-drop framework to a 2D torus setting, providing a rigorous foundation for rotating drops and their symmetry-related multiplicity, with potential implications for understanding capillary-driven dynamics on curved manifolds. The analysis relies on a detailed treatment of the Dirichlet-Neumann operator and conformal geometry, yielding precise variational and Hamiltonian characterizations of the rotating waves.

Abstract

We consider the free boundary problem for a two-dimensional, incompressible, perfect, irrotational liquid drop of nearly circular shape with capillarity: that is, we consider the 2D version of the 3D capillary drop problem treated in Baldi-Julin-La Manna [11] and Baldi-La Manna-La Scala [12]. In particular, we derive its Craig-Sulem formulation firstly over the circle, then over the one-dimensional flat torus; the arising equations are similar to the pure capillary Water Waves for the ocean problem, apart from conformal factors and additional terms due to curvature terms. Then, we show its Hamiltonian structure and we derive constants of motions from symmetries, one of which is the invariance by the torus action. Thanks to this invariance, we show the existence of orbits of rotating wave solutions (which are the analogous of travelling waves of the ocean problem) by bifurcation from multiple eigenvalues in the spirit of Moser-Weinstein [44, 56] and Craig-Nicholls [22] variational approaches; in particular, we can parametrize such orbits by the angular momentum, and for each value of it they are unique. This will imply that each orbit is generated by symmetric rotating waves.
Paper Structure (12 sections, 27 theorems, 256 equations)

This paper contains 12 sections, 27 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric objects on surfaces and their representation on the infinite periodic strip
  4. The spaces of Sobolev and analytic functions on ${\mathbb T}^1$
  5. The capillary drop equations
  6. The Craig-Sulem formulation on ${\mathbb S}^1$ and ${\mathbb T}^1$
  7. Hamiltonian structure, symmetries and invariants of motion
  8. Rotating waves
  9. The rotating wave equations
  10. The linearized operator at static circle solution
  11. Existence of rotating waves
  12. Existence of symmetric rotating waves

Key Result

Lemma 2.1

Let $\Omega_h$ be as in Om.h, and let $\gamma_h$ be as in gamma.diff. For any function $f\colon\, \partial \Omega_h \to {\mathbb R}$, let $\tilde{f}$ be its pullback by $\gamma_h$, namely $\tilde{f}(x) = f\circ\gamma_h(x))$. Then

Theorems & Definitions (54)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4: from LannesLannes1BMV
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 44 more