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Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves

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Pith's one-line read This paper constructs small-amplitude, linearly stable, time quasi-periodic traveling wave solutions to the 3D pure-gravity water wave equations in finite depth, for generic lattices and almost every depth.

desk verdict A genuine first-result KAM paper for 3D gravity water waves; the proof is coherent and detailed in the visible sections, but the load-bearing measure-theoretic core (Prop. 6.4) is unverified in the text I saw, and that is where a specialist referee must dig. read the letter →

arxiv 2509.10318 v1 pith:YRBXOBWH submitted 2025-09-12 math.AP math-phmath.MP

Roberto Feola , Riccardo Montalto , Shulamit Terracina This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35Q3535B4035S0576B15
keywords quasi-periodictravelingwavespuregravitywaterfinitedepthKAMtheorysmalldivisorsDirichlet-NeumannoperatornormalformsMelnikovconditions
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

The paper aims to prove that the three-dimensional pure-gravity water wave equations on a flat torus, in finite depth, admit global-in-time solutions whose motion is genuinely quasi-periodic in time and traveling in space, not reducible to stationary Stokes waves in any moving frame. The existence holds for a generic choice of the horizontal lattice, for any finite number of propagation speeds, and for an asymptotically full-measure set of depths as the amplitude shrinks. The solutions are small, arbitrarily smooth, and linearly stable. If the proof is right, this supplies the first non-steady global solutions for 3D water waves on compact domains and the first KAM-type result for an autonomous quasi-linear dispersive PDE in dimension greater than one.

What carries the argument

The key machinery is a normal-form reduction of the linearized water wave operator at a quasi-periodic traveling wave. It combines a quantitative Egorov theorem and sharp tame pseudo-differential estimates for the Dirichlet-Neumann operator, the use of the sublinear dispersion relation to let the time derivative ω·∂_φ dominate the order-1/2 spatial operator, and conservation of momentum, which enforces that momentum-preserving time-independent operators are Fourier multipliers. This first stage reduces the linearized operator to a bounded Fourier multiplier up to smoothing remainders; a second KAM stage diagonalizes it using weak Melnikov conditions, with losses controlled by the smoothness

What would settle it

At the unperturbed level, for a fixed finite set of tangential sites, compute the zeros in h of the second-Melnikov functions ω(h)·ℓ + Ω(j;h) ± Ω(j';h) under the momentum constraint; if any non-zero triple (ℓ,j,j') yields a zero set containing an interval, or if h ↦ that combination has zero derivative at a zero, then the transversality behind Proposition 6.4 breaks and the full-measure depth estimate would fail.

Watch

Extended reading notes

Core claim

On the paper's own terms, Theorem 1.4 states that, under Hypothesis 1.2 on the rational independence of the dual-lattice Gram matrix, for any finite set of tangential wave vectors with distinct lengths and for a large-measure set of depths h, the pure-gravity water wave system has a small-amplitude quasi-periodic traveling wave solution with a Diophantine frequency vector ω=ω(h,ζ) close to the linear frequencies, whose remainder is o(√ζ) in high Sobolev regularity. The central discovery is that this can be achieved despite three compounding difficulties: the weak sublinear dispersion relation √(|j|tanh(h|j|)), the quasi-linear (derivative-order) nonlinearity, and the absence of a sharp asymp

Load-bearing premise

The construction needs the depth parameter h to separate the resonances: although the linear frequencies depend on h only through exponentially small corrections, those corrections—together with the rational independence of the lattice Gram matrix and momentum conservation—must make every relevant Melnikov combination transverse in h; if that fails, the set of usable depths need not have full measure and the iteration cannot run.

Editorial extensions

If this is right

  • The constructed quasi-periodic tori are global-in-time and linearly stable: the linearized flow near each torus is conjugate to a diagonal operator with purely imaginary spectrum in the normal directions.
  • The solutions are not stationary in any moving reference frame, so they enlarge the known building blocks of 3D periodic water waves beyond traveling Stokes waves.
  • The tori exist for an asymptotically full-measure set of depths as the amplitude tends to zero, with Diophantine frequencies that adjust with the amplitude.
  • The solutions are arbitrarily Sobolev regular in both time and space, so they are classical smooth solutions for high enough regularity indices.
  • The two-stage reducibility strategy, built on sublinear dispersion and momentum conservation, is presented as a template for other translation-invariant quasi-linear dispersive PDEs in dimension at least two.

Reading between the lines

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

  • If the transversality mechanism fails for lattices not satisfying Hypothesis 1.2—for example, a square lattice with rational Gram-matrix components—then the asymptotically full-measure depth set may collapse; this is a natural numerical and theoretical boundary to probe.
  • The same normal-form scheme might carry over to other fluid models with sublinear dispersion and conserved momenta, such as internal waves or rotating shallow water, provided the linearized operator admits the same pseudo-differential structure.
  • Because the proof avoids reversibility assumptions, imposing reversibility as an additional symmetry would likely yield reversible (standing-type) quasi-periodic waves in 3D as well, a case the paper does not explicitly state.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper claims Theorem 1.4: under a generic non-resonance condition on the dual lattice (Hypothesis 1.2), for any finite ordered set of tangential wave vectors and for a Cantor-like set of depths with asymptotically full measure, the 3D pure-gravity water-wave equations on a flat torus admit small-amplitude, time quasi-periodic traveling solutions with an arbitrary number of rationally independent speeds. These solutions are asserted to be global in time, not stationary in any moving frame, and linearly stable. The strategy is a Nash-Moser/KAM scheme built on a detailed pseudo-differential expansion of the Dirichlet-Neumann operator, a reduction of the linearized operator to constant coefficients up to smoothing remainders, and a KAM diagonalization controlled by second Melnikov conditions and conservation of momentum. The version under review contains substantial parts of the functional calculus and the elliptic estimates in Sections 2-4, but the decisive Sections 5-13, including Proposition 6.4 and the measure estimates, are not reproduced and are only summarized in the introduction.

Significance. If the proof is correct, this is a major advance: it would be the first KAM result for an autonomous, quasi-linear, dispersive PDE in dimension greater than one, and the first construction of global time-quasi-periodic solutions of 3D water waves on compact domains that are not steady in any moving frame. The paper has clear strengths: it formulates an explicit genericity condition on the lattice, develops a substantial space-time tame pseudo-differential calculus for the Dirichlet-Neumann operator, and uses momentum conservation in a genuinely non-trivial way to remove exact resonances. I found no circularity in the strategy: the non-resonance conditions are imposed on parameter sets whose measure is claimed to be large, not assumed to equal the conclusion. However, the central measure statement rests on Proposition 6.4, which is not available in the submitted text; the proof of the theorem therefore cannot currently be certified.

major comments (3)
  1. [§6 / Prop. 6.4] The theorem's 'asymptotically full measure' claim for the good set G_ζ depends on transversality properties of the unperturbed Melnikov functions. The introduction, after (1.17), states that |∂_h^k r(j,h)| ≤ C_k e^{-h|j|}, and says the required transversality is proved in Proposition 6.4 using momentum conservation and Hypothesis 1.2. In the version under review, Section 6 and Proposition 6.4 are not reproduced. This is load-bearing: for F_{ℓ,j,j'}(h)=ω(h)·ℓ+ω_j(h)-ω_{j'}(h) with V^Tℓ+j-j'=0, the derivative is ℓ·ω'(h)+∂_h(ω_j-ω_j'). The second term is O(⟨j⟩^{3/2} e^{-2h|j|}), while ℓ·ω'(h) is a linear form in ℓ with fixed positive coefficients; for unfavorable h it can be superpolynomially small even when |ℓ|≍|j|. A merely exponential lower bound on |∂_h F| would give bad-set measure O(γ e^{c|j|}), whose sum over j,j' diverges. The proof must supply a polynomial, or at least summable, lo
  2. [§12 and §13.2] The set G_∞ is defined in (13.37) as the intersection over n of Λ_{γ_n}^∞(i~_n), i.e. infinitely many 0th-, 1st-, and 2nd-order Melnikov conditions at every Nash-Moser step. The measure estimate for such an intersection is not a formality. The eigenvalues μ_∞(j;λ,i) are explicitly said not to have an asymptotic expansion in powers of 1/|j|, and the only quantitative handle stated is |∇m_7(j)| ≲ ⟨j⟩^{-1/2} in Lemma 13.5. In Section 12, the second Melnikov condition (1.54) has an extra factor |j'|^τ in the denominator, and Lemma 12.4 is asserted to control the resulting derivative loss using the smoothing structure of the remainder and momentum conservation. These sections are not present in the version under review. Since the Nash-Moser inversion (Theorem 8.5) depends on the invertibility estimate (8.19), this is a second load-bearing gap in the proof as submitted.
  3. [§5 / Theorem 4.2] The entire normal-form reduction starts from the pseudo-differential expansion of G(η), Theorem 4.2. The proof is deferred to Section 5, whose content is not included in the version under review. The lemmas in Section 4 provide elliptic estimates but do not by themselves establish the symbol expansion (1.30) with remainders satisfying the unbalanced tame estimates (1.32). These remainders are later used to compensate for the derivative losses from the weak Melnikov conditions. As the submitted text stands, this foundational step is unverified, and the subsequent reducibility argument cannot be checked from the material provided.
minor comments (5)
  1. [Hypothesis 4.1] Typo: 'bounded iopen set' should read 'bounded open set'.
  2. [Theorem 1.4] The quantifier over the tangential sites S is not precise. It is not stated whether ε0(s) and the measure estimates are allowed to depend on |ȷ_i|; if they do, the phrase 'for any choice of wave vectors' should specify that dependence, since the h-derivatives of the linear frequencies decay exponentially in |ȷ_i|.
  3. [Lemma 2.8] The composition estimate (2.35) has different derivative losses in the two terms (s+k+1 and s0+k+2). If intentional, this should be explained; otherwise it may be a typo.
  4. [§2.4 / (2.89)-(2.91)] The symbol χ is used both for the frequency cut-off in (2.89) and for the cut-off in (2.39). This is confusing and should be renamed.
  5. [§5 / Lemma 5.8] The discussion around (1.42)-(1.43) refers to |Re(a)| ≳ |ξ| to make y^M e^{y a} a symbol of order -M, but the sign convention on the symbol a is only described in words. Please state the hypothesis explicitly before the lemma that uses it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the KAM/Nash-Moser construction and measure estimates are self-contained; non-resonance conditions are proven, not fitted.

full rationale

The paper's central claim is a Nash-Moser/KAM existence theorem for quasi-periodic traveling gravity water waves. The derivation is not circular: the non-resonance (Diophantine and Melnikov) conditions are imposed on the parameter set G_∞, and the paper proves both (i) that these conditions imply invertibility of the linearized operator via an elaborate reducibility scheme, and (ii) that the excluded set of parameters is small in measure, using Hypothesis 1.2 and momentum conservation, with the key transversality statement formulated as Proposition 6.4. The frequency vector ω is a free parameter chosen in an open set, and the counterterms α, μ in (7.6) are adjusted by the Nash-Moser/inverse-function mechanism; they are not fitted to the target conclusion. The amplitude parameter ζ is an input measuring the size of the torus, not a parameter calibrated to force existence. The theorem's conclusion (existence for h in a large-measure Cantor-like set) is not equivalent to the definition of G_∞: G_∞ is defined by sufficient small-divisor conditions whose measure is proven large, and the proof of that measure statement is a nontrivial part of the paper. Self-citations, e.g., to [5], [27], and [40], occur for technical lemmas and general strategy, but the core pseudo-differential expansion, Egorov estimates, symmetrization, block-diagonalization, KAM reducibility, and measure estimates are developed in the present paper. The skeptic's concern about Proposition 6.4 is a concern about whether the transversality estimate is correct and verified, not a circular reduction of the conclusion to its assumptions. No step has been exhibited where an input is renamed as a prediction or where a load-bearing premise reduces to a self-citation.

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

The proof rests on standard Sobolev and pseudo-differential calculus, the Hamiltonian formulation of water waves, Nash-Moser/KAM machinery, and the paper's own Hypothesis 1.2 on the lattice. No free parameters are fitted to data; the amplitude, frequencies, and depths are parameters of the theorem. No new entities are postulated.

assumptions (6)
  • domain assumption Hypothesis 1.2: the Gram matrix vector P=(Kj·Kk) has rationally independent components.
    Assumed on the dual lattice; used to rule out exact resonances between tangential and normal modes and to prove measure estimates.
  • domain assumption Finite depth h in [h1,h2] with 0<h1<h2.
    The theorem is stated for finite depth; depth is the parameter used to impose non-resonance conditions.
  • standard math The water waves system is Hamiltonian with conserved momenta, and the linearized operator preserves momentum.
    Zakharov/Craig-Sulem formulation (1.6)-(1.10); used throughout the normal form reduction.
  • standard math Standard properties of Sobolev spaces, pseudo-differential operators, and the Dirichlet-Neumann operator estimates used as building blocks.
    Background functional analysis and PDE results; many are reproved with tame estimates in sections 2 to 5.
  • domain assumption The dispersion relation is sub-linear: Omega(xi)=|xi|^{1/2} tanh(h|xi|)^{1/2}, so the time derivative term dominates at highest order.
    The whole reduction scheme exploits sub-linearity; the authors explicitly exclude the super-linear capillary case.
  • domain assumption Diophantine and Melnikov non-resonance conditions hold on a large-measure parameter set.
    The KAM iteration requires these conditions; sections 6 and 13.2 are devoted to proving they hold on a set of asymptotically full measure.

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Cite this review

Pith. "Pith review of Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves." pith.science (2026). https://pith.science/paper/YRBXOBWH

@misc{pith2026250910318,
  author       = {Pith},
  title        = {Pith review of: Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRBXOBWH}},
  note         = {Machine review of arXiv:2509.10318}
}
read the original abstract

Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics (which is a widely open problem). In this paper we prove the existence of time quasi-periodic traveling wave solutions for three-dimensional pure gravity water waves in finite depth, on flat tori, with an arbitrary number of speeds of propagation. These solutions are global in time, they do not reduce to stationary solutions in any moving reference frame and they are approximately given by finite sums of Stokes waves traveling with rationally independent speeds of propagation. This is a very hard small divisors problem for Partial Differential Equations due to the fact that one deals with a dispersive quasi-linear PDE in higher dimension with a very complicated geometry of the resonances. Our result is the first KAM (Kolmogorov-Arnold-Moser) result for an autonomous, dispersive, quasi-linear PDE in dimension greater than one and it is the first example of global solutions, which do not reduce to steady ones in any moving reference frame, for 3D water waves equations on compact domains.

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Forward citations

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Reference graph

Works this paper leans on

94 extracted references · 7 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Alazard T., Baldi P.,Gravity capillary standing water waves, Arch. Rat. Mech. Anal, 217:741–830, (2015)

    2015

  2. [2]

    374, (2015)

    Alazard T., Delort J-M.,Sobolev estimates for two dimensional gravity water waves, Ast ´erisque, vol. 374, (2015)

    2015

  3. [3]

    Alazard T., Delort J-M.Global solutions and asymptotic behavior for two dimensional gravity water waves. Ann. Sci. ´Ec. Norm. Sup´er., 48(5):1149–1238, (2015)

    2015

  4. [4]

    Alazard T., M ´etivier G.,Paralinearization of the Dirichlet to Neumann operator, and regularity of the three dimen- sional water waves. Comm. Partial Differential Equations 34, no. 10-12, 1632–1704, (2009)

    2009

  5. [5]

    214(2):739–911, (2018)

    Baldi P., Berti M., Haus E., Montalto R.,Time quasi-periodic gravity water waves in finite depth, Inventiones Math. 214(2):739–911, (2018)

    2018

  6. [6]

    Baldi P., Berti M., Montalto R.,KAM for autonomous quasi-linear perturbations of KdV, Ann. I. H. Poincar ´e (C) Anal. Non Lin´eaire, 33(6):1589–1638, (2015)

    2015

  7. [7]

    Baldi P., Berti M., Montalto R.,KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann. 359:471–536, (2014)

    2014

  8. [8]

    Baldi P., Montalto R.,Quasi-periodic incompressible Euler flows in 3D, Advances in Mathematics, 384:107730, (2021)

    2021

Show all 94 references
  1. [9]

    Bambusi, D., Gr ´ebert B., Maspero A., Robert D.,Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent perturbation, Analysis & PDE, 11(3):775–799, (2018)

    2018

  2. [10]

    Henri Poincar´e 20(6):1893–1929, (2019)

    Bambusi D., Langella B., Montalto R.,Reducibility of non-resonant transport equation onT d with unbounded perturbations, Ann. Henri Poincar´e 20(6):1893–1929, (2019)

    1929

  3. [11]

    Communications in Partial differential equations 45(4), pp

    Bambusi D., Langella B., Montalto R.,On the spectrum of the Schr¨odinger operator onT d: a normal form approach. Communications in Partial differential equations 45(4), pp. 303-320 (2020)

    2020

  4. [12]

    Bambusi D., Langella B., Montalto R.,Spectral asymptotics of all the eigenvalues of Schr ¨odinger operators on flat tori.Nonlinear Analysis, Theory, Methods and Applications, 216, 112679 (2022)

    2022

  5. [13]

    344–358 (2022)

    Bambusi D., Langella B., Montalto R.,Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori.Journal of Differential Equations, 318, pp. 344–358 (2022)

    2022

  6. [14]

    de L’Ecole Normale

    Bambusi D., Langella B.,Growth of Sobolev norms in quasi integrable quantum systems.To appear in Ann. de L’Ecole Normale. Preprint arXiv:2202.04505 (2022)

    2022 arXiv

  7. [15]

    Benyamini Y ., Lindenstrauss J.,Geometric nonlinear functional analysis. Vol. 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, (2000)

    2000

  8. [16]

    255-284, (2015)

    Berti M., Bolle P.,A Nash-Moser approach to KAM theory, Fields Institute Communications, special volume ”Hamil- tonian PDEs and Applications”, pp. 255-284, (2015)

    2015

  9. [17]

    Berti M., Bolle P.,Quasi-Periodic Solutions of Nonlinear Wave Equations on thed-Dimensional Torus, EMS Series of Lectures in Math. (2020)

    2020

  10. [18]

    UMI Lecture Notes 2018 (awarded UMI book prize 2017), ISBN 978-3-319-99486-4

    Berti M., Delort J.-M.,Almost Global Solutions of Capillary-gravity Water Waves Equations on the Circle. UMI Lecture Notes 2018 (awarded UMI book prize 2017), ISBN 978-3-319-99486-4

    2018

  11. [19]

    Pure and Applied Math

    Berti M., Feola R., Pusateri f.,Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves, Comm. Pure and Applied Math. 76(7):1416–1494, (2023)

    2023

  12. [20]

    Berti M., Feola R., Franzoi L.,Quadratic life span of periodic gravity-capillary water waves, Water Waves, 3:85– 115, (2021)

    2021

  13. [21]

    Berti M., Feola R., Procesi M., Terracina S.,Reducibility of Klein-Gordon equations with maximal order perturba- tions, to appear on Memoirs of the European Mathematical Society, preprint arXiv:2402.11377

    arXiv

  14. [22]

    Rational Mech

    Berti M., Franzoi L., Maspero A.,Traveling quasi-periodic water waves with constant vorticity, Arch. Rational Mech. Anal., 240:99–202, (2021)

    2021

  15. [23]

    Pure Appl

    Berti M., Franzoi L., Maspero A.,Pure gravity traveling quasi-periodic water waves with constant vorticity, Comm. Pure Appl. Math. 77(3):990–1064, (2024)

    2024

  16. [24]

    Berti M., Hassainia Z., Masmoudi N.,Time quasi-periodic vortex patches of Euler equation in the plane. Invent. Math. 233: 1279–1391, (2023)

    2023

  17. [25]

    Berti M., Maspero A., Murgante F.,Hamiltonian Birkhoff normal form for gravity-capillary water waves with con- stant vorticity: almost global existence.Annals of PDEs 10, 22 (2024)

    2024

  18. [26]

    Berti M., Maspero A., Ventura P., On the analyticity of the Dirichlet–Neumann operator and Stokes waves.Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.33: (3) 611-650, (2022)

    2022

  19. [27]

    263 (2020)

    Berti M., Montalto R.,Quasi-periodic standing wave solutions of gravity-capillary water waves, Memoires of AMS, Memo 891, vol. 263 (2020)

    2020

  20. [28]

    Berti M., Kappeler T., Montalto R.,Large KAM Tori for Quasi-linear Perturbations of KdV, Arch Rational Mech Anal 239, 1395–1500 (2021)

    2021

  21. [29]

    Bianchini R., Franzoi L., Montalto R., Terracina S.,Large Amplitude Quasi-Periodic Traveling Waves in Two Di- mensional Forced Rotating Fluids, Commun. Math. Phys. , 406(66), (2025). 204

    2025

  22. [30]

    Ciampa G., Montalto R., Terracina S.,Large amplitude traveling waves for the non-resistive MHD system,Journal of Hyperbolic Differential Equations, 21(3):707–790, (2024)

    2024

  23. [31]

    SIAM, (2011)

    Constantin A.,Nonlinear Water Waves with Applications to Wave-Current Interaction and Tsunamis, CBMS-NSF Regional Conf, Series in Applied Math., 81. SIAM, (2011)

    2011

  24. [32]

    Constantin A., Ivanov R.I., Prodanov E.M.,Nearly-Hamiltonian structure for water waves with constant vorticity, J. Math. Fluid Mech. 10, 224-237, (2008)

    2008

  25. [33]

    Pure Appl

    Constantin A., Strauss W.,Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57, no. 4, 481-527, (2004)

    2004

  26. [34]

    Craig W., Nicholls D.,Travelling two and three dimensional capillary gravity water waves,SIAM J. Math. Anal., 32(2):323-359, (2000)

    2000

  27. [35]

    Craig W., Sulem C.,Numerical simulation of gravity waves, J. Comput. Phys., 108(1):73–83, (1993)

    1993

  28. [36]

    Dubreil-Jacotin M.-L.,Sur la d ´etermination rigoureuse des ondes permanentes p´eriodiques d’ampleur finie, J. Math. Pures Appl. 13, 217-291, (1934)

    1934

  29. [37]

    H., Gr ´ebert B., Kuksin S

    Eliasson L. H., Gr ´ebert B., Kuksin S. B.,KAM for the nonlinear beam equation, Geom. Funct. Anal. 26 : 1588–715, (2016)

    2016

  30. [38]

    H., Kuksin S

    Eliasson L. H., Kuksin S. B.,KAM for the nonlinear Schr ¨odinger equation, Ann. Math. (2) 172, no. 1 : 371–435, (2010)

    2010

  31. [39]

    Journal Diff

    Enciso A., Peralta-Salas D., Torres de Lizaur F.,Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher. Journal Diff. Eq. 354, 170-182 (2023)

    2023

  32. [40]

    295, num

    Feola, Giuliani F.,Quasi-periodic traveling waves on an infinitely deep prefect fluid under gravity, Memoires of AMS, vol. 295, num. 1471, (2024)

    2024

  33. [41]

    Feola R., Giuliani F., Montalto R., Procesi M.,Reducibility of first order linear operators on tori via Moser’s theorem, Dynamics of partial differential equations, 16(1), 25-94 (2019)

    2019

  34. [42]

    Feola R., Giuliani F., Procesi M.,Reducibility for a class of weakly dispersive linear operators arising from the Degasperis–Procesi equation, Dynamics of partial differential equations, 16(1), 25-94 (2019)

    2019

  35. [43]

    Feola R., Giuliani F., Procesi M.,Reducibile KAM tori for the Degasperis-Procesi equation, Commun. Math. Phys. 377, 1681–1759 (2020)

    2020

  36. [44]

    Feola R., Gr ´ebert B.,Reducibility of Schr¨odinger equation on the sphere, Int. math. Res. Not. 2021:19, 15082–15120, (2021)

    2021

  37. [45]

    Feola R., Iandoli F.,Local well-posedness for quasi-linear NLS with large Cauchy data on the circle, Ann. I. H. Poincar´e (C) Anal. Non Lin´eaire, 36(1):119-164, (2019)

    2019

  38. [46]

    Feola R., Iandoli F.,Long time existence for fully nonlinear NLS with small Cauchy data on the circle, Annali Scuola Normale Superiore - classe di scienze, 22(1), 109-182. (2021)

    2021

  39. [47]

    Archive for Rational Mechanics and Analysis (ARMA), V ol

    Franzoi L., Masmoudi N., Montalto R.Space Quasi-Periodic Steady Euler Flows Close to the Inviscid Couette Flow. Archive for Rational Mechanics and Analysis (ARMA), V ol. 248, no. 81, (2024)

    2024

  40. [48]

    Annales Henri Poincar´e, V ol

    Franzoi L., Montalto R.,A KAM approach to the inviscid limit for the 2D Navier-Stokes equations. Annales Henri Poincar´e, V ol. 25, No. 12, 5231–5275, (2024)

    2024

  41. [49]

    Mathematics in Engineering, 6(3): 394–406, (2024)

    Franzoi L., Montalto R.,Time almost-periodic solutions of the incompressible Euler equations. Mathematics in Engineering, 6(3): 394–406, (2024)

    2024

  42. [50]

    Geng J., Jiangong You J.,A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces, Commun. Math. Phys. 262, 343–372, (2006)

    2006

  43. [51]

    Germain P., Masmoudi N., Shatah J.,Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), 175(2):691–754, (2012)

    2012

  44. [52]

    K ¨onigl

    Gerstner F.,Theorie der Wellen, Abh. K ¨onigl. B¨ohm. Ges. Wiss, (1802)

  45. [53]

    G ´omez-Serrano J., Ionescu A.D., Park J.,Quasiperiodic solutions of the generalized SQG equation, preprint arxiv:2303.03992, (2023)

    2023 arXiv

  46. [54]

    Goyon R.,Contribution ´a la th´eorie des houles, Ann. Sci. Univ. Toulouse 22, 1-55, (1958)

    1958

  47. [55]

    Journal of Differential Equations 413: 129–189, (2024)

    Groves M.D., Nilsson D., Pasquali S., Wahl´en E.,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

  48. [56]

    Hassainia Z., Hmidi T., Masmoudi N.,KAM theory for active scalar equations, to appear in Memoirs of the American Mathematical Society, arxiv:2110.08615, (2021)

    2021 arXiv

  49. [57]

    Hassainia Z., Hmidi T., Masmoudi N.,Rigorous derivation of the leapfrogging motion for planar Euler equations, preprinthttps://arxiv.org/abs/2311.15765

    arXiv

  50. [58]

    Hassainia Z., Hmidi T., Roulley E.,Desingularization of time-periodic vortex motion in bounded domains via KAM tools, preprinthttps://arxiv.org/abs/2408.16671

    arXiv

  51. [59]

    Hassainia Z., Hmidi T., Roulley E.,Invariant KAM tori around annular vortex patches for 2D Euler equations, Communications in Mathematical Physics, 405, 270, (2024). 205

    2024

  52. [60]

    preprint, arxiv:2110.13751

    Hmidi T., Roulley E.,Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations. preprint, arxiv:2110.13751

    arXiv

  53. [61]

    Hunter J., Ifrim M., Tataru D.,Two dimensional water waves in holomorphic coordinates.Comm. Math. Phys. 346, 483-552, (2016)

    2016

  54. [62]

    Ifrim M., Tataru D.,Two dimensional water waves in holomorphic coordinates II: global solutions, Bull. Soc. Math. France 144(2):369–394, (2016)

    2016

  55. [63]

    Math., 199(3):653–804, (2015)

    Ionescu A., Pusateri F.,Global solutions for the gravity water waves system in 2d, Invent. Math., 199(3):653–804, (2015)

    2015

  56. [64]

    Ionescu A., Pusateri F.,Long-time existence for multi-dimensional periodic water waves, Geom. Funct. Anal. 29(3):811–870, (2019)

    2019

  57. [65]

    Iooss G., Plotnikov P.,Small divisor problem in the theory of three-dimensional water gravity waves, Mem. Amer. Math. Soc., 200(940):viii+128, (2009)

    2009

  58. [66]

    Iooss G., Plotnikov P.,Asymmetrical tridimensional traveling gravity waves, Arch. Rat. Mech. Anal., 200(3):789– 880, (2011)

    2011

  59. [67]

    Iooss G., Plotnikov P., Toland J.,Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal., 177(3), 367-478, (2005)

    2005

  60. [68]

    Keady G., Norbury J.,On the existence theory for irrotational water waves,Math. Proc. Cambridge Philos. Soc. 83, no. 1, 137-157, (1978)

    1978

  61. [69]

    Lannes D.,Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18(3):605–654, (2005)

    2005

  62. [70]

    Mathematical Surveys and Mono- graphs, 188, (2013)

    Lannes D.,The water waves problem: mathematical analysis and asymptotics. Mathematical Surveys and Mono- graphs, 188, (2013)

    2013

  63. [71]

    Levi-Civita T.,D ´etermination rigoureuse des ondes permanentes d’ampleur finie. Math. Ann., 93 , pp. 264-314, (1925)

    1925

  64. [72]

    Lions J.L., Magenes E.,Non-Homogeneous Boundary Value Problems and Applications, V ol. 1. Springer, Berlin- Heidelberg-New York, (1972). https://doi.org/10.1007/978-3-642-65161-8

    1972 doi

  65. [73]

    Real World Appl

    Martin C.,Local bifurcation and regularity for steady periodic capillary-gravity water waves with constant vorticity, Nonlinear Anal. Real World Appl. 14, no. 1, 131-149, (2013)

    2013

  66. [74]

    M ´etivier G.,Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Publications of the Scuola Normale Superiore (2008)

    2008

  67. [75]

    Montalto R.,The Navier-Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi- Periodic Solutions. J. Dyn. Differ. Equations 33, 1341-1362 (2021)

    2021

  68. [76]

    Montalto R.,A reducibility result for a class of linear wave equations onT d, Int. Math. Res. Notices 2019, no. 6: 1788–862, (2019)

    2019

  69. [77]

    Moser.A rapidly convergent iteration method and non-linear partial differential equations - I

    J. Moser.A rapidly convergent iteration method and non-linear partial differential equations - I. Ann. Scuola Norm. Sup. di Pisa - Classe Scienze, 20(2): 265-315, (1966)

    1966

  70. [78]

    Plotnikov P., Toland J.,Nash-Moser theory for standing water waves, Arch. Ration. Mech. Anal., 159(1):1–83, (2001)

    2001

  71. [79]

    Procesi, C. and M. Procesi.A KAM algorithm for the non-linear Schr ¨odinger equation, Adv. Math. 272: 399–470, (2015)

    2015

  72. [80]

    Dynamics of Partial Differential Equations 20, no

    Roulley E.,Periodic and quasi-periodic Euler-alpha flows close to Rankine vortices. Dynamics of Partial Differential Equations 20, no. 4, 311-366 (2023)

    2023

  73. [81]

    Chaotic Dyn

    R ¨ussmann H.,Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn. 6(2), 199-204, (2001)

    2001

  74. [82]

    Advances in Mathematics 447,109683, (2024)

    Seth D.S., Varholm K., Wahl´en E.,Symmetric doubly periodic gravity-capillary waves with small vorticity. Advances in Mathematics 447,109683, (2024)

    2024

  75. [83]

    Princeton Mathematical Series, no

    Stein E.M.,Singular integrals and differentiability properties of functions. Princeton Mathematical Series, no. 30. Princeton University Press, Princeton, N.J. (1970)

    1970

  76. [84]

    Cambridge Phil

    Stokes G.,On the theory of oscillatory waves, Trans. Cambridge Phil. Soc. 8, 441-455, (1847)

  77. [85]

    Ann., 95: 595–634, (1926)

    Struik D.,D ´etermination rigoureuse des ondes irrotationnelles p´eriodiques dans un canal `a profondeur finie, Math. Ann., 95: 595–634, (1926)

    1926

  78. [86]

    F.,On the existence of a wave of greatest height and Stokes conjecture, Proc

    Toland J. F.,On the existence of a wave of greatest height and Stokes conjecture, Proc. Roy. Soc. London Ser. A 363, 1715, 469-485, (1978)

    1978

  79. [87]

    Wahl ´en E.,Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal. 38, 921-943, (2006)

    2006

  80. [88]

    Physics, 79, 303-315, (2007)

    Wahl ´en E.,A Hamiltonian formulation of water waves with constant vorticity, Letters in Math. Physics, 79, 303-315, (2007)

    2007

  81. [89]

    Wahl ´en E., Weber J.,Large-amplitude steady gravity water waves with general vorticity and critical layers.Duke Math. J. 173(11): 2197-2258, (2024)

    2024

  82. [90]

    Taylor,Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, Birkh ¨auser, (1991)

    M.E. Taylor,Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, Birkh ¨auser, (1991). 206

    1991

  83. [91]

    Wu S.,Well-posedness in Sobolev spaces of the full water waves problem in 2-D. Invent. Math. 130(1), 39-72, (1997)

    1997

  84. [92]

    Math., 184(1):125–220, (2011)

    Wu S.,Global well-posedness of the 3-D full water wave problem, Invent. Math., 184(1):125–220, (2011)

    2011

  85. [93]

    Zakharov V .E.,Stability of periodic waves of finite amplitude on the surface of a deep fluid, J Appl Mech Tech Phys 9:190–194, (1968)

    1968

  86. [94]

    Ra- tion

    Zeidler E.,Existenzbeweis f ´ur permanente Kapillar-Schwerewellen mit allgemeinen Wirbelverteilungen, Arch. Ra- tion. Mech. Anal., 50, 34–72, (1973). DIPARTIMENTO DIMATEMATICA EFISICA, UNIVERSIT `A DEGLISTUDIROMATRE, LARGOSANLEONARDOMURIALDO1, 00146, ROMAITALY Email address:ro...

    1973

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