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Two-dimensional water waves with constant vorticity and general bottom topography

S. Pasquali

TL;DR

This work extends the theory of two-dimensional gravity-capillary water waves to include constant vorticity in finite-depth fluids with general bottom topography. It introduces a generalized Dirichlet-Neumann-type operator $G(\eta,\beta,\gamma)$, proves its analyticity and a detailed paralinearization, and derives a homogeneous expansion around flat surface as well as a robust para-differential decomposition. Using these tools, the authors establish a local well-posedness result for the water waves system in Sobolev spaces, accommodating less regular bottoms and velocities than prior irrotational results. The methods unify and extend Zakharov-Craig-Sulem-type formulations to rotational flows, providing a rigorous framework for gravity-capillary waves with variable bottom and constant vorticity. Overall, the paper delivers new analytic and PDE techniques for studying 2D water waves with vorticity and nontrivial bottom topography, with potential implications for accurate oceanographic modeling and wave-structure interactions.

Abstract

In this paper we consider two-dimensional water waves with constant vorticity, under the action of gravity and surface tension, in a fluid domain with finite depth and general bottom topography. We present a formulation which generalizes the one by Zakharov-Craig-Sulem for irrotational water waves, and the one by Constantin-Ivanov-Prodanov for water waves with constant vorticity and flat bottom topography. We study in detail an operator which appears in such formulation, extending well-known results for the classical Dirichlet-Neumann operator, such as an analiticity result, the Taylor expansion in homogeneous powers of the wave profile, and a paralinearization formula. As an application, we prove a local well-posedness result.

Two-dimensional water waves with constant vorticity and general bottom topography

TL;DR

This work extends the theory of two-dimensional gravity-capillary water waves to include constant vorticity in finite-depth fluids with general bottom topography. It introduces a generalized Dirichlet-Neumann-type operator , proves its analyticity and a detailed paralinearization, and derives a homogeneous expansion around flat surface as well as a robust para-differential decomposition. Using these tools, the authors establish a local well-posedness result for the water waves system in Sobolev spaces, accommodating less regular bottoms and velocities than prior irrotational results. The methods unify and extend Zakharov-Craig-Sulem-type formulations to rotational flows, providing a rigorous framework for gravity-capillary waves with variable bottom and constant vorticity. Overall, the paper delivers new analytic and PDE techniques for studying 2D water waves with vorticity and nontrivial bottom topography, with potential implications for accurate oceanographic modeling and wave-structure interactions.

Abstract

In this paper we consider two-dimensional water waves with constant vorticity, under the action of gravity and surface tension, in a fluid domain with finite depth and general bottom topography. We present a formulation which generalizes the one by Zakharov-Craig-Sulem for irrotational water waves, and the one by Constantin-Ivanov-Prodanov for water waves with constant vorticity and flat bottom topography. We study in detail an operator which appears in such formulation, extending well-known results for the classical Dirichlet-Neumann operator, such as an analiticity result, the Taylor expansion in homogeneous powers of the wave profile, and a paralinearization formula. As an application, we prove a local well-posedness result.
Paper Structure (20 sections, 51 theorems, 275 equations)

This paper contains 20 sections, 51 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Formulation for the flat bottom case
  4. Formulation for general bottom topography
  5. Main results
  6. Properties of the operator $G(\eta,\beta,\gamma)$
  7. Local well-posedness
  8. Properties of the operator $G(\eta,\beta,\gamma)$
  9. Analiticity
  10. Homogeneous expansion around $\eta=0$
  11. Paralinearization of the operator $G(\eta,\beta,\gamma)$
  12. Proof of Proposition \ref{['prop:OpGpara']}
  13. Local well-posedness
  14. Paralinearization of the water waves system
  15. Symmetrization
  16. ...and 5 more sections

Key Result

Theorem 1

Let $s > 3/2$, and fix $\psi \in H^{s}(\mathbb{T})$, $\gamma \in \mathbb{R}$. If we consider the operator $G(\eta,\beta,\gamma)$ defined in eq:OpG, then the map is analytic.

Theorems & Definitions (87)

  • Remark 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • ...and 77 more