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On the gravity-capillary water waves with point vortices

Lizhe Wan

TL;DR

This paper advances the theory of two-dimensional gravity-capillary water waves by incorporating point vortices into the full Cauchy problem and reformulating the system in holomorphic coordinates. It develops an energy framework and establishes local well-posedness for regularity $s>\frac{3}{2}$, together with a robust a priori estimate that couples the free surface dynamics to finite-velocity point vortices through a Helmholtz–Kirchhoff model. For the two-vortex symmetric setting, it introduces a normal-form reduction and a carefully designed modified energy to prove an extended cubic lifespan for small data, highlighting how rotational effects can be tamed in this regime. The analysis employs paradifferential calculus and holomorphic projection to control nonlinear interactions, both on the surface and in the bulk, while tracking the vortex dynamics via ODEs coupled to the free boundary evolution. Overall, the work extends local well-posedness and long-time behavior results to gravity-capillary waves with point vortices, bridging irrotational and rotational fluid models in a rigorous PDE-ODE framework.

Abstract

We consider the two-dimensional deep gravity-capillary water waves with point vortices. We first formulate the question in the holomorphic coordinates. Then, we derive an a priori energy estimate for water waves, and show that the water wave system has a unique solution for initial data in $\mathcal{H}^s\times Ω_t^N$, $s>\frac{3}{2}$. Finally, we show that if there are only two vortices, and vortices, initial velocity, and initial surfaces are symmetric with respect to the vertical axis, then the solution of pure capillary water waves with two point vortices has an extended cubic lifespan.

On the gravity-capillary water waves with point vortices

TL;DR

This paper advances the theory of two-dimensional gravity-capillary water waves by incorporating point vortices into the full Cauchy problem and reformulating the system in holomorphic coordinates. It develops an energy framework and establishes local well-posedness for regularity , together with a robust a priori estimate that couples the free surface dynamics to finite-velocity point vortices through a Helmholtz–Kirchhoff model. For the two-vortex symmetric setting, it introduces a normal-form reduction and a carefully designed modified energy to prove an extended cubic lifespan for small data, highlighting how rotational effects can be tamed in this regime. The analysis employs paradifferential calculus and holomorphic projection to control nonlinear interactions, both on the surface and in the bulk, while tracking the vortex dynamics via ODEs coupled to the free boundary evolution. Overall, the work extends local well-posedness and long-time behavior results to gravity-capillary waves with point vortices, bridging irrotational and rotational fluid models in a rigorous PDE-ODE framework.

Abstract

We consider the two-dimensional deep gravity-capillary water waves with point vortices. We first formulate the question in the holomorphic coordinates. Then, we derive an a priori energy estimate for water waves, and show that the water wave system has a unique solution for initial data in , . Finally, we show that if there are only two vortices, and vortices, initial velocity, and initial surfaces are symmetric with respect to the vertical axis, then the solution of pure capillary water waves with two point vortices has an extended cubic lifespan.

Paper Structure

This paper contains 19 sections, 28 theorems, 283 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Water waves in holomorphic coordinates
  3. Main results
  4. The organization of the paper
  5. Preliminaries and derivation of water waves
  6. Norms and function spaces
  7. Paradifferential estimates
  8. Derivation of water waves in holomorphic coordinates
  9. Water waves related estimates
  10. Local well-posedness of water waves
  11. A priori energy estimate for water waves
  12. The proof of local well-posedness
  13. Cubic lifespan of solutions for 2 vortices with small symmetrized data
  14. Preliminary estimates
  15. Normal form variables
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $s>0$, suppose $({\mathbf W}, R, \{z_j\}_{j=1}^N)\in C( [0,T]; {\mathcal{H} }^s)\times C^1([0,T];\Omega_t)^N$ solve the water waves e:WW. Then, there exists an energy functional $E_s$ which has the following properties:

Figures (1)

  • Figure 1: The conformal mapping of the domain

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2: MR3052498
  • Lemma 2.3: MR3585049
  • Lemma 2.4: MR2768550
  • Lemma 2.5: MR2768550
  • ...and 35 more