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Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity

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Pith's one-line read This paper proves that the generalized Dirichlet–Neumann operator for three-dimensional water waves with vorticity is analytic in the surface profile, gives its recursive Taylor coefficients, and derives its paralinearization.

desk verdict Solid toolkit extension: arbitrary-vorticity DNO results are believable and mostly explicit, but the analyticity proof leans on an analogy to [GH20] that is not written out, and the abstract overstates the Taylor expansion by hiding the gamma0 = 0 condition. read the letter →

arxiv 2502.09370 v2 pith:EOIWLAVZ submitted 2025-02-13 math.AP

S. Pasquali This is my paper · ORCID
classification math.AP MSC 37K4576B0376B15
keywords waterwavesvorticityDirichlet–NeumannoperatorZakharov–Craig–SulemformulationanalyticityTaylorexpansionparalinearization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the boundary-value description of three-dimensional gravity water waves when the flow carries vorticity, using a generalized Zakharov–Craig–Sulem formulation. Its aim is to establish that the associated generalized Dirichlet–Neumann operator, which sends boundary data to the normal velocity at the free surface, behaves like the classical operator of irrotational water-wave theory: it is analytic in the surface elevation, admits an explicit Taylor expansion around the flat state, and has a paralinearization. Under regularity and smallness assumptions on the surface profile, the boundary velocity potential, and the straightened vorticity, the author proves these results without imposing any geometric condition on the velocity field or on the vorticity. This would give rotational flows the same boundary calculus that supports perturbation, long-time, and asymptotic analyses for irrotational waves. The recursive Taylor formulas for the vorticity part are stated under the additional condition $\gamma_0=0$, where $\gamma$ is the vorticity-dependent quantity defined in Proposition 2.3.

What carries the argument

The load-bearing object is the generalized Dirichlet–Neumann operator $G_{\mathrm{gen}}[\eta](\Phi,\omega)=U\cdot N|_{z=\eta}$, defined through the boundary-value problem $\mathrm{curl}\,U=\omega$, $\mathrm{div}\,U=0$, with bottom condition and prescribed tangential trace at the surface. The paper flattens the domain with a straightening diffeomorphism $\Sigma$, rewrites the div-curl system as a flattened elliptic problem, and represents its solution through an explicit Green matrix built from $\sinh(|\xi|(\zeta+h))$ and $\cosh(|\xi|w)$. Analyticity is obtained by proving that the solution operators depend analytically on $\sigma$ and linearly on the data; the Taylor coefficients are generated by differentiating the differential formula (6.15) and evaluating at $\delta\eta=\eta$. The paralinearization uses a localizing transform to a strip, a good unknown $\hat B=\hat A - T_{\partial_w^\varrho \hat A}\eta$ that absorbs the leading surface terms, and a factorization of the flattened elliptic operator into paradifferential first-order factors with strongly elliptic principal symbols.

What would settle it

Computing the degree-zero homogeneous term $\gamma_0$ of $\partial_w^\Sigma \tilde{\omega}\, d\sigma(\eta)$ for a regularizing diffeomorphism and a shear vorticity would settle the matter: a nonzero value for data satisfying the paper's hypotheses disproves the unconditional Taylor expansion, and a zero value in all such cases shows the condition is redundant.

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Extended reading notes

Core claim

The central claim is analyticity: for $s\ge 2$, with $\tilde{\omega}\in (H^{s-2}(D_0))^3$, $\Phi\in \dot H^{s-1/2}(\mathbb{R}^2)$, and a straightening diffeomorphism satisfying (2.5), the maps $\tilde G_{\mathrm{gen,I}}[\eta]$ and $\tilde G_{\mathrm{gen,II}}[\eta]$ are analytic in $\eta$ near 0 as operators on these data. The zeroth-order terms are explicit: $\tilde G_{0,\mathrm{I}}\Phi = \mathcal{F}^{-1}[\,|\xi|\tanh(h|\xi|)\,\mathcal{F}\Phi\,]$, and $\tilde G_{0,\mathrm{II}}\tilde{\omega}$ is given by the Green-matrix formula (6.13) involving an integral in depth of the horizontal vorticity and a boundary term. For $j\ge 1$, if $\tilde{\gamma}_0=0$, the homogeneous coefficients satisfy the recursions (6.14) and (6.19). The paper also derives the differential of the rotational part, $d\tilde G_{\mathrm{gen,II}}[\eta](\delta\eta)\tilde{\omega} = -\tilde G_{\mathrm{gen,II}}[\eta]\tilde{\gamma} - \nabla\cdot[(\tilde K_{\mathrm{II}}[\eta]\tilde{\omega} - \tilde W_{\mathrm{II}}[\eta]\tilde{\omega}\nabla\eta)\delta\eta]$ with $\tilde{\gamma}=\partial_w^\Sigma\tilde{\omega}\,d\sigma(\delta\eta)$, and proves a paralinearization formula (7.3) with remainder in $H^{2n-3-\varepsilon}$.

Load-bearing premise

The recursive Taylor expansion of the vorticity part is proven only when the degree-zero homogeneous term $\gamma_0 = (\partial_w^\Sigma \tilde{\omega}\, d\sigma)|_{\delta\eta=\eta}^0$ vanishes, and this condition depends jointly on the vorticity and on the chosen straightening diffeomorphism rather than following from the paper's stated regularity and smallness assumptions.

Editorial extensions

If this is right

  • The classical perturbation toolkit for irrotational water waves—analytic dependence on the profile, Taylor coefficients, differentials, paralinearization—now extends to flows with arbitrary vorticity, so local bifurcation and long-time analyses can be based on a boundary formulation.
  • The explicit zeroth-order operator separates the irrotational symbol $|\xi|\tanh(h|\xi|)$ from a depth-integrated vorticity term, so the effect of vorticity at the flat surface is a computable correction at every order of the amplitude expansion.
  • The differential formula provides the linearization of the rotational part around any small profile, a basic ingredient for linearized and weakly nonlinear analyses.
  • The paralinearization formula (7.3) gives a quasilinear representation of the vorticity contribution, suitable for energy estimates in Sobolev spaces.
  • The velocity-field estimates give a priori control of the full three-dimensional flow in terms of the surface data and the straightened vorticity, which can serve as the control needed in well-posedness and continuation arguments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If $\tilde{\gamma}_0$ fails to vanish for some admissible data, the homogeneous Taylor expansion of the rotational part is incomplete; a natural follow-up would be to choose the straightening diffeomorphism so that $\tilde{\gamma}_0$ is absorbed into the zeroth-order operator, restoring an unconditional expansion.
  • Because every order of the expansion is generated by explicit Green-matrix integrals, the recursive coefficients could be evaluated numerically for shear or Beltrami-type vorticities, giving a practical check of the expansion.
  • The same Green-matrix flattening strategy could be adapted to related free-boundary models with density stratification or surface tension, where a generalized Dirichlet–Neumann operator is less developed.
  • A concrete consequence of the paralinearization is that the vorticity part should exhibit the same leading-order paradifferential structure as the irrotational part; comparing the symbol $\lambda_{\mathrm{II}}$ with direct numerical solutions for a shear flow would test the formula.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies the generalized Dirichlet–Neumann operator arising in the Castro–Lannes formulation of three-dimensional water waves with arbitrary vorticity. After introducing a straightening diffeomorphism, it proves higher-order estimates for the velocity field (Theorem 1/5), analyticity of the straightened operator with respect to the surface elevation (Theorem 2/Corollary 6.2), a formula for the differential of the rotational part (Propositions 2.3 and 6.3), a homogeneous Taylor expansion with recursive coefficient formulas (Theorems 3 and 7; equations (6.12)–(6.14) and (6.18)–(6.19)), and a paralinearization formula for the rotational part (Theorems 4 and 8). The advertised feature is that no geometric condition on the velocity field or vorticity is assumed.

Significance. If fully substantiated, these results would provide a useful and quite explicit toolkit for rotational water-wave problems, extending the irrotational theory and the recent Beltrami-flow work in [GNPW24]. The Green-matrix representation in Lemma 6.1 and the recursive coefficient formulas are concrete and likely to be reusable. The paper is generally careful with function spaces and gives a self-contained proof of Proposition 1.3 in Appendix A. However, the significance is currently tempered by two load-bearing gaps: the central analyticity theorem is delegated to an analogous argument in [GH20], and the Taylor expansion is proved only under an additional condition γ0 = 0 that is not part of the advertised hypotheses. A further regularity hypothesis enters the paralinearization theorem without being derived.

major comments (3)
  1. [Section 6, Theorem 6] The main analyticity theorem is not proved in the manuscript. Immediately after Lemma 6.1 the text states that 'Following an argument analogous to the proof of Theorem 4.10 in [GH20]' one obtains Theorem 6, but the remainder terms in the flattened system (6.2)–(6.6) depend on the arbitrary straightened vorticity in a nonlinear way. For instance, r31[σ, \tilde A, \tildeω] defined after (6.6) contains -(curlΣ \tilde A)_h|0 + (curl \tilde A)_h|0 - (curlΣ \tilde A)_3|0 ∇η + ∇^⊥ Δ^{-1}(\tildeω(·,0)·∇η). To conclude that (σ, \tildeω) ↦ \tilde A is analytic, one must show that all these remainder maps are analytic on the stated Sobolev spaces and that the fixed-point equation supplied by the Green matrix in Lemma 6.1 is locally uniquely solvable with analytic dependence on the data. That verification is not written out. This is a gap in support, not a demonstrated contradiction, but it is load-bearing: Theorem 2 and the Taylor formulas of Theorem 7 inherit their analyticity from Theorem 6. The same mode of deferral appears in Corollary 4.3, proved by analogy to Proposition 4.8 of [GH20], and in Propositions A.3–A.4.
  2. [Section 6, Theorem 7] The homogeneous Taylor expansion is established only under the condition \tildeγ0 = 0, and this condition is not shown to follow from the regularity, smallness, and condition (2.5). The text acknowledges that \tildeγ0 = 0 depends on both the vorticity and the choice of straightening diffeomorphism and gives only examples where it holds, such as the trivial diffeomorphism with vorticity near the origin. Consequently, the explicit formulas (6.18) and (6.19) are not established for general data with \tildeγ0 ≠ 0. The abstract's announcement of a Taylor expansion with 'no geometric condition' therefore overstates the proved statement. The theorem should either prove \tildeγ0 = 0 under a transparent hypothesis, or state the expansion with this condition explicitly and adjust the advertised scope.
  3. [Section 7, Theorem 8] The paralinearization formula (7.3) is conditional on the a priori regularity assumption (7.2) on \hat A and \hatω. The paper does not show how (7.2) follows from the hypotheses on η and \hatω; Theorem 6 only provides \tilde A ∈ (H^s(D0))^3, which does not by itself imply the conditions ∂^k_w \hat A ∈ C0([-h,0]; H^{n-k}(R2)) for k = 0,1 when n may be larger than s. Without such a derivation, the statement is an identity for functions \hat A that happen to satisfy (7.2), rather than a theorem about \hat G_gen,II[η]\hatω on a natural space. Please either prove the needed regularity from the boundary value problem or state explicitly that the paralinearization theorem is conditional on (7.2).
minor comments (3)
  1. [Sections 2 and 5] The statement called Theorem 1 in Section 2.1 reappears as Theorem 5 in Section 5, and Theorem 4 reappears as Theorem 8 in Section 7. Please use a single numbering scheme for the restated theorems.
  2. [Section 1, equation (1.29)] The displayed equation for the irrotational ZCS system ends with '= 0.,(1.29)'; the stray period before the equation number should be removed.
  3. [Section 6, before Theorem 7] The quantity \tildeγ0 is central to the validity of the Taylor expansion, but it is introduced only in a paragraph and not in a numbered equation. A formal definition of \tildeγ0 and its dependence on \tildeω and σ would help the reader verify the condition and its scope.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Taylor expansion, differential formula, and paralinearization are derived from explicit boundary value problems and Green matrices; delegated proofs are support gaps, not circular inputs.

full rationale

The paper's central claims do not reduce to their inputs by construction. The analyticity result (Theorem 2) and the Taylor coefficients (Theorems 3 and 7) are obtained by flattening the boundary value problems (6.2)-(6.9), solving them with the explicit Green matrices of Lemma 6.1, and reading off the zeroth-order terms (6.12)-(6.13); the higher-order coefficients are then given by the recursion formulas (6.14) and (6.18)-(6.19) obtained from (6.11) and from the differential formula (6.15) proved in Proposition 6.3. No parameter is fitted to the quantity being 'predicted'. The only notable delegation is Theorem 6, whose proof is described as 'Following an argument analogous to the proof of Theorem 4.10 in [GH20]'; since the remainders r31 etc. couple an arbitrary straightened vorticity, this is an unverified support gap for the analyticity step, but it is not a circular reduction because the cited result is an external theorem for Beltrami flows rather than the conclusion being derived. Similarly, formula (6.14) cites the author's prior work GNPW24, but it is the classical Craig-Sulem recursion and is also stated to be deducible from (6.11), so the self-citation is not load-bearing. The extra hypothesis gamma0=0 is explicitly stated as a condition depending on vorticity and the chosen straightening diffeomorphism, not smuggled in. The paralinearization (Theorem 8) is proved in Section 7 with paradifferential calculus and elliptic estimates following Alazard-Metivier. Thus no circular step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters and no new physical or mathematical entities. The central derivations rely on standard PDE and paradifferential background, on the Castro-Lannes formulation, and on one ad hoc compatibility condition gamma0 = 0 that limits the Taylor expansion theorem.

assumptions (6)
  • domain assumption The div-curl boundary value problem (1.18) and (3.3) has a unique solution with the stated H1 estimates (Proposition 1.3 and Proposition 3.2).
    Adopted as the starting point of the analysis. The proof in Appendix A relies on standard elliptic theory and on arguments analogous to [CL15] and [GH20], partly via citation.
  • domain assumption There exists a regularizing straightening diffeomorphism sigma satisfying the control conditions (2.4) or (2.5).
    Proposition 1.9 constructs such a diffeomorphism for small enough delta, and the main theorems assume its existence. This is a structural hypothesis on the parametrization, not a consequence of the target results.
  • domain assumption The generalized Zakharov-Craig-Sulem formulation of Castro and Lannes [CL15] is valid and reduces to the classical Zakharov-Craig-Sulem formulation when the vorticity vanishes.
    This formulation is used throughout as the basis for defining the generalized Dirichlet-Neumann operator and for deriving all the main results.
  • standard math Standard paradifferential calculus rules hold, including symbolic calculus, product estimates, commutator estimates, and the elliptic regularity result in Proposition B.14.
    Quoted from Metivier, Alazard-Delort, Alazard-Burq-Zuily, and Taylor; these are background tools used in Section 7 and Appendices.
  • standard math Sobolev, Zygmund, and Holder embeddings hold in the function spaces used, including Hs(R2) subset C^{s-1}(R2) and Hs(D0) subset C^{s-3/2}(D0).
    Used to justify trace and regularity thresholds throughout the paper, following standard references such as Taylor and Bahouri-Chemin-Danchin.
  • ad hoc to paper The condition gamma0 = 0, where gamma := partial^Sigma_w tilde-omega d sigma evaluated at delta eta = eta, holds for the Taylor expansion in Theorem 3 and Theorem 7.
    This condition is not guaranteed by the no-geometric-condition assumptions. The paper notes that it holds in the irrotational case and for small vorticity with the trivial diffeomorphism, but it is a nontrivial restriction on the main expansion theorem.

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Pith. "Pith review of Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity." pith.science (2026). https://pith.science/paper/EOIWLAVZ

@misc{pith2026250209370,
  author       = {Pith},
  title        = {Pith review of: Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOIWLAVZ}},
  note         = {Machine review of arXiv:2502.09370}
}
read the original abstract

In this paper we consider three-dimensional water waves with vorticity, under the action of gravity. We discuss a generalized Zakharov-Craig-Sulem formulation of the problem introduced by Castro and Lannes, which involves a generalized Dirichlet-Neumann operator. We study this operator in detail, extending some well-known results about the classical Dirichlet-Neumann operator for irrotational water waves, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and a paralinearization result. We stress the fact that no geometric condition on either the velocity field or the vorticity is assumed.

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Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · cited by 1 Pith paper

  1. [1]

    Gravity capillary standing water waves

    Thomas Alazard and Pietro Baldi. Gravity capillary standing water waves . Archive for Rational Mechanics and Analysis , 217(3):741--830, 2015

    work page 2015

  2. [2]

    On the water-wave equations with surface tension

    Thomas Alazard, Nicolas Burq, and Claude Zuily. On the water-wave equations with surface tension . Duke Mathematical Journal , 158(3):413--499, 2011

    work page 2011

  3. [3]

    On the Cauchy problem for gravity water waves

    Thomas Alazard, Nicolas Burq, and Claude Zuily. On the Cauchy problem for gravity water waves . Inventiones mathematicae , 198(1):71--163, 2014

    work page 2014

  4. [4]

    Cauchy theory for the gravity water waves system with non-localized initial data

    Thomas Alazard, Nicolas Burq, and Claude Zuily. Cauchy theory for the gravity water waves system with non-localized initial data . Annales de l'Institut Henri Poincar \'e C, Analyse non lin \'e aire , 33(2):337--395, 2016

    work page 2016

  5. [5]

    Sobolev estimates for two dimensional gravity water waves

    Thomas Alazard and Jean-Marc Delort. Sobolev estimates for two dimensional gravity water waves . Ast\'erisque , 374:viii--241, 2015

    work page 2015

  6. [6]

    Two-dimensional gravity waves at low regularity II: Global solutions

    Albert Ai, Mihaela Ifrim, and Daniel Tataru. Two-dimensional gravity waves at low regularity II: Global solutions . Annales de l'Institut Henri Poincar \'e C , 39(4):819--884, 2022

    work page 2022

  7. [7]

    Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves

    Thomas Alazard and Guy M \'e tivier. Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves . Communications in Partial Differential Equations , 34(12):1632--1704, 2009

    work page 2009

  8. [8]

    Hamiltonian studies on counter-propagating water waves

    Dario Bambusi. Hamiltonian studies on counter-propagating water waves . Water Waves , 3(1):49--83, 2021

    work page 2021

Show all 62 references
  1. [9]

    Time quasi-periodic gravity water waves in finite depth

    Pietro Baldi, Massimiliano Berti, Emanuele Haus, and Riccardo Montalto. Time quasi-periodic gravity water waves in finite depth . Inventiones mathematicae , 214(2):739--911, 2018

    2018

  2. [10]

    Fourier analysis and nonlinear partial differential equations

    Hajer Bahouri, Jean-Yves Chemin, and Rapha \"e l Danchin. Fourier analysis and nonlinear partial differential equations . Springer, 2011

    2011

  3. [11]

    Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle

    Massimiliano Berti and Jean-Marc Delort. Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle . Springer, 2018

    2018

  4. [12]

    Impulse, flow force and variational principles

    T Brooke Benjamin. Impulse, flow force and variational principles . IMA Journal of applied Mathematics , 32(1-3):3--68, 1984

    1984

  5. [13]

    Traveling quasi-periodic water waves with constant vorticity

    Massimiliano Berti, Luca Franzoi, and Alberto Maspero. Traveling quasi-periodic water waves with constant vorticity . Archive for Rational Mechanics and Analysis , 240(1):99--202, 2021

    2021

  6. [14]

    Pure gravity traveling quasi-periodic water waves with constant vorticity

    Massimiliano Berti, Luca Franzoi, and Alberto Maspero. Pure gravity traveling quasi-periodic water waves with constant vorticity . Communications on Pure and Applied Mathematics , 77(2):990--1064, 2024

    2024

  7. [15]

    A variational reduction and the existence of a fully localised solitary wave for the three-dimensional water-wave problem with weak surface tension

    Boris Buffoni, Mark D Groves, and Erik Wahl \'e n. A variational reduction and the existence of a fully localised solitary wave for the three-dimensional water-wave problem with weak surface tension . Archive for Rational Mechanics and Analysis , 228:773--820, 2018

    2018

  8. [16]

    Fully localised three-dimensional gravity-capillary solitary waves on water of infinite depth

    Boris Buffoni, Mark D Groves, and Erik Wahl \'e n. Fully localised three-dimensional gravity-capillary solitary waves on water of infinite depth . Journal of Mathematical Fluid Mechanics , 24(2):55, 2022

    2022

  9. [17]

    Hamiltonian long-wave approximations to the water-wave problem

    Walter Craig and Mark D Groves. Hamiltonian long-wave approximations to the water-wave problem . Wave motion , 19(4):367--389, 1994

    1994

  10. [18]

    On the motion of the free surface of a liquid

    Demetrios Christodoulou and Hans Lindblad. On the motion of the free surface of a liquid . Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences , 53(12):1536--1602, 2000

    2000

  11. [19]

    Well-posedness and shallow-water stability for a new Hamiltonian formulation of the water waves equations with vorticity

    Angel Castro and David Lannes. Well-posedness and shallow-water stability for a new Hamiltonian formulation of the water waves equations with vorticity . Indiana University Mathematics Journal , pages 1169--1270, 2015

    2015

  12. [20]

    Traveling two and three dimensional capillary gravity water waves

    Walter Craig and David P Nicholls. Traveling two and three dimensional capillary gravity water waves . SIAM Journal on Mathematical Analysis , 32(2):323--359, 2000

    2000

  13. [21]

    Traveling gravity water waves in two and three dimensions

    Walter Craig and David P Nicholls. Traveling gravity water waves in two and three dimensions . European Journal of Mechanics-B/Fluids , 21(6):615--641, 2002

    2002

  14. [22]

    Numerical simulation of gravity waves

    Walter Craig and Catherine Sulem. Numerical simulation of gravity waves . Journal of computational physics , 108(1):73--83, 1993

    1993

  15. [23]

    Exact steady periodic water waves with vorticity

    Adrian Constantin and Walter Strauss. Exact steady periodic water waves with vorticity . Communications on Pure and Applied Mathematics , 57(4):481--527, 2004

    2004

  16. [24]

    Well-posedness of the free-surface incompressible Euler equations with or without surface tension

    Daniel Coutand and Steve Shkoller. Well-posedness of the free-surface incompressible Euler equations with or without surface tension . Journal of the American Mathematical Society , 20(3):829--930, 2007

    2007

  17. [25]

    Global solutions of the gravity-capillary water-wave system in three dimensions

    Yu Deng, Alexandru D Ionescu, Beno \^ t Pausader, and Fabio Pusateri. Global solutions of the gravity-capillary water-wave system in three dimensions. Acta Mathematica , 219:213–--402, 2017

    2017

  18. [26]

    Smooth stationary water waves with exponentially localized vorticity

    Mats Ehrnstr \"o m, Samuel Walsh, and Chongchun Zeng. Smooth stationary water waves with exponentially localized vorticity . Journal of the European Mathematical Society , 25(3):1045--1090, 2022

    2022

  19. [27]

    Quasi-periodic traveling waves on an infinitely deep perfect fluid under gravity , volume 295

    Roberto Feola and Filippo Giuliani. Quasi-periodic traveling waves on an infinitely deep perfect fluid under gravity , volume 295. American Mathematical Society, 2024

    2024

  20. [28]

    Theorie der wellen

    Franz Gerstner. Theorie der wellen . Annalen der Physik , 32(8):412--445, 1809

  21. [29]

    A variational formulation for steady surface water waves on a Beltrami flow

    Mark D Groves and J Horn. A variational formulation for steady surface water waves on a Beltrami flow . Proceedings of the Royal Society A , 476(2234):20190495, 2020

    2020

  22. [30]

    Global solutions for the gravity water waves equation in dimension 3

    Pierre Germain, Nader Masmoudi, and Jalal Shatah. Global solutions for the gravity water waves equation in dimension 3 . Annals of Mathematics , pages 691--754, 2012

    2012

  23. [31]

    Analytical study of a generalised Dirichlet--Neumann operator and application to three-dimensional water waves on Beltrami flows

    Mark D Groves, Dag Nilsson, Stefano Pasquali, and Erik Wahl \'e n. Analytical study of a generalised Dirichlet--Neumann operator and application to three-dimensional water waves on Beltrami flows . Journal of Differential Equations , 413:129--189, 2024

    2024

  24. [32]

    Long time regularity for 3d gravity waves with vorticity

    Daniel Ginsberg and Fabio Pusateri. Long time regularity for 3d gravity waves with vorticity . arXiv preprint arXiv:2401.10096 , 2024

    2024 arXiv

  25. [33]

    Traveling water waves--the ebb and flow of two centuries

    Susanna Haziot, Vera Hur, Walter Strauss, John Toland, Erik Wahl \'e n, Samuel Walsh, and Miles Wheeler. Traveling water waves--the ebb and flow of two centuries . Quarterly of applied mathematics , 80(2):317--401, 2022

    2022

  26. [34]

    Lectures on nonlinear hyperbolic differential equations , volume 26

    Lars H \"o rmander. Lectures on nonlinear hyperbolic differential equations , volume 26. Springer Science & Business Media, 1997

    1997

  27. [35]

    Small divisor problem in the theory of three-dimensional water gravity waves

    G \'e rard Iooss and Pavel I Plotnikov. Small divisor problem in the theory of three-dimensional water gravity waves . American Mathematical Soc., 2009

    2009

  28. [36]

    Asymmetrical three-dimensional travelling gravity waves

    G \'e rard Iooss and Pavel Plotnikov. Asymmetrical three-dimensional travelling gravity waves . Archive for rational mechanics and analysis , 200(3):789--880, 2011

    2011

  29. [37]

    Global solutions for the gravity water waves system in 2d

    Alexandru D Ionescu and Fabio Pusateri. Global solutions for the gravity water waves system in 2d . Inventiones mathematicae , 199:653--804, 2015

    2015

  30. [38]

    Long-time existence for multi-dimensional periodic water waves

    Alexandru D Ionescu and Fabio Pusateri. Long-time existence for multi-dimensional periodic water waves . Geometric and Functional Analysis , 29(3):811--870, 2019

    2019

  31. [39]

    Standing waves on an infinitely deep perfect fluid under gravity

    G \'e rard Iooss, Pavel I Plotnikov, and John F Toland. Standing waves on an infinitely deep perfect fluid under gravity . Archive for rational mechanics and analysis , 177(3):367--478, 2005

    2005

  32. [40]

    Sharp hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary euler equations

    Mihaela Ifrim, Ben Pineau, Daniel Tataru, and Mitchell A Taylor. Sharp hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary euler equations. Annals of PDE , 11(1):16, 2025

    2025

  33. [41]

    Two-dimensional gravity water waves with constant vorticity, I: Cubic lifespan

    Mihaela Ifrim and Daniel Tataru. Two-dimensional gravity water waves with constant vorticity, I: Cubic lifespan . Analysis & PDE , 12(4):903--967, 2018

    2018

  34. [42]

    Well-posedness of the water-waves equations

    David Lannes. Well-posedness of the water-waves equations . Journal of the American Mathematical Society , 18(3):605--654, 2005

    2005

  35. [43]

    The water waves problem: mathematical analysis and asymptotics , volume 188

    David Lannes. The water waves problem: mathematical analysis and asymptotics , volume 188. American Mathematical Soc., 2013

    2013

  36. [44]

    D \'e termination rigoureuse des ondes permanentes d'ampleur finie

    Tullio Levi-Civita. D \'e termination rigoureuse des ondes permanentes d'ampleur finie . Mathematische Annalen , 93(1):264--314, 1925

    1925

  37. [45]

    Well-posedness for the motion of an incompressible liquid with free surface boundary

    Hans Lindblad. Well-posedness for the motion of an incompressible liquid with free surface boundary . Annals of mathematics , pages 109--194, 2005

    2005

  38. [46]

    An Existence Theory for Small-Amplitude Doubly Periodic Water Waves with Vorticity

    Evgeniy Lokharu, DS Seth, and E Wahl \'e n. An Existence Theory for Small-Amplitude Doubly Periodic Water Waves with Vorticity . Archive for Rational Mechanics and Analysis , 238(2):607--637, 2020

    2020

  39. [47]

    A variational principle for three-dimensional water waves over Beltrami flows

    Evgeniy Lokharu and Erik Wahl \'e n. A variational principle for three-dimensional water waves over Beltrami flows . Nonlinear Analysis , 184:193--209, 2019

    2019

  40. [48]

    Paradifferential Calculus and Application to the Cauchy Problem for Nonlinear Systems

    Guy M \'e tivier. Paradifferential Calculus and Application to the Cauchy Problem for Nonlinear Systems . CRM Series, Edizioni della Scuola Normale Superiore , 2008

    2008

  41. [49]

    A new approach to analyticity of Dirichlet-Neumann operators

    David P Nicholls and Fernando Reitich. A new approach to analyticity of Dirichlet-Neumann operators . Proceedings of the Royal Society of Edinburgh Section A: Mathematics , 131(6):1411--1433, 2001

    2001

  42. [50]

    Three-dimensional, nonlinear wave interaction in water of constant depth

    John Reeder and Marvin Shinbrot. Three-dimensional, nonlinear wave interaction in water of constant depth . Nonlinear Analysis: Theory, Methods & Applications , 5(3):303--323, 1981

    1981

  43. [51]

    Pseudodifferential operators and spectral theory , volume 57

    Mikhail Aleksandrovich Shubin. Pseudodifferential operators and spectral theory , volume 57. Springer, 2001

    2001

  44. [52]

    On the theory of oscillatory waves

    George Gabriel Stokes. On the theory of oscillatory waves . Transactions of the Cambridge philosophical society , 1880

  45. [53]

    Symmetric doubly periodic gravity-capillary waves with small vorticity

    Douglas S Seth, Kristoffer Varholm, and Erik Wahl \'e n. Symmetric doubly periodic gravity-capillary waves with small vorticity . Advances in Mathematics , 447:109683, 2024

    2024

  46. [54]

    Local well-posedness for fluid interface problems

    Jalal Shatah and Chongchun Zeng. Local well-posedness for fluid interface problems . Archive for rational mechanics and analysis , 199(2):653--705, 2011

    2011

  47. [55]

    Partial differential equations III: Nonlinear Equations , volume 117

    Michael Taylor. Partial differential equations III: Nonlinear Equations , volume 117. Springer Science & Business Media, 2011

    2011

  48. [56]

    Commutator estimates for H \"o lder continuous and bmo-Sobolev multipliers

    Michael Taylor. Commutator estimates for H \"o lder continuous and bmo-Sobolev multipliers . Proceedings of the American Mathematical Society , 143(12):5265--5274, 2015

    2015

  49. [57]

    Steady periodic capillary-gravity waves with vorticity

    Erik Wahl \'e n. Steady periodic capillary-gravity waves with vorticity . SIAM journal on mathematical analysis , 38(3):921--943, 2006

    2006

  50. [58]

    A Hamiltonian formulation of water waves with constant vorticity

    Erik Wahl \'e n. A Hamiltonian formulation of water waves with constant vorticity . Letters in Mathematical Physics , 79(3):303--315, 2007

    2007

  51. [59]

    Almost global wellposedness of the 2-D full water wave problem

    Sijue Wu. Almost global wellposedness of the 2-D full water wave problem . Inventiones mathematicae , 177(1):45--135, 2009

    2009

  52. [60]

    Global wellposedness of the 3-D full water wave problem

    Sijue Wu. Global wellposedness of the 3-D full water wave problem . Inventiones mathematicae , 184(1):125--220, 2011

    2011

  53. [61]

    Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity

    Erik Wahl \'e n and J \"o rg Weber. Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity . International Mathematics Research Notices , 2023(20):17377--17410, 2023

    2023

  54. [62]

    Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary

    Chao Wang, Zhifei Zhang, Weiren Zhao, and Yunrui Zheng. Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary . Memoirs of the American Mathematical Society , 270(1318), 2021

    2021

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