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Equivariant critical point theory and bifurcation of $3d$ gravity-capillary Stokes waves

Tommaso Barbieri, Massimiliano Berti, Marco Mazzucchelli

Abstract

We establish novel existence results of $3d$ gravity-capillary periodic traveling waves. In particular we prove the bifurcation of multiple, geometrically distinct truly $3d$ Stokes waves having the same momentum of any non-resonant $2d$ Stokes wave. This unexpected clustering phenomenon of Stokes waves, observed in physical fluids, is a fundamental consequence of the Hamiltonian nature of the water waves equations, their symmetry groups, and novel topological arguments. We employ a variational Lyapunov-Schmidt reduction combined with equivariant Morse-Conley theory for a functional defined on a joined topological space invariant under a $2$-torus action. Although the reduction is a priori singular near the hyperplanes of $2d$-waves, we circumvent this difficulty by exhaustive use of the symmetry groups. This approach yields a complete bifurcation picture of $3d $ gravity-capillary Stokes waves.

Equivariant critical point theory and bifurcation of $3d$ gravity-capillary Stokes waves

Abstract

We establish novel existence results of gravity-capillary periodic traveling waves. In particular we prove the bifurcation of multiple, geometrically distinct truly Stokes waves having the same momentum of any non-resonant Stokes wave. This unexpected clustering phenomenon of Stokes waves, observed in physical fluids, is a fundamental consequence of the Hamiltonian nature of the water waves equations, their symmetry groups, and novel topological arguments. We employ a variational Lyapunov-Schmidt reduction combined with equivariant Morse-Conley theory for a functional defined on a joined topological space invariant under a -torus action. Although the reduction is a priori singular near the hyperplanes of -waves, we circumvent this difficulty by exhaustive use of the symmetry groups. This approach yields a complete bifurcation picture of gravity-capillary Stokes waves.

Paper Structure

This paper contains 46 sections, 71 theorems, 381 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Main results
  4. Ideas of the proof
  5. Equivariant critical point theory
  6. Modules and algebras.
  7. Gradings.
  8. Equivariant cohomology
  9. Classifying spaces.
  10. Borel quotient.
  11. Equivariant cohomology.
  12. Orbits of the action
  13. Annihilator of the equivariant cohomology
  14. The Euler class and the Gysin sequence
  15. Some computations of equivariant cohomology
  16. ...and 31 more sections

Key Result

Theorem 1.1

(Multiplicity of $3d$-Stokes waves: collinear case) Assume that the number of resonant wave vectors is $\# {\mathcal{V}} \geq 3$ (resonant case). Then there is $\varepsilon >0$ such that there exist, in addition to a unique $S^1$-orbit of ${2d}$ Stokes wave $u_*$ along the direction $j_0$ with momentum ${\mathcal{I}}(u_*) = a$, at least $N := \#{\mathcal{V}}-{2}$ geometrically distinct truly $3d$

Theorems & Definitions (153)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Lemma 1.5
  • proof
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • ...and 143 more