Pith. sign in

REVIEW 2 major objections 2 minor 34 references

Small scale creation in 2D gravity-capillary water waves with vorticity

T0 review · 2 major / 2 minor · reviewed 2026-05-10 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Initial data with a flat free surface and small velocity can produce at least double-exponential growth in the L^∞ norm of the vorticity gradient for 2D gravity-capillary water waves.

desk verdict The paper gives an explicit construction for double-exponential growth of vorticity gradient in 2D gravity-capillary free-surface flow, generalizing Zlatos and Hu-Luo-Yao, but the control on existence time is the part that needs checking. read the letter →

arxiv 2604.19380 v1 submitted 2026-04-21 math.AP

Yuanpeng Tu This is my paper · ORCID
classification math.AP
keywords 2Dwaterwavesgravity-capillaryvorticitygradientdouble-exponentialgrowthfreesurfacesmallscalecreationEulerequationsunboundeddomain
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 constructs specific initial data for the 2D incompressible Euler equations in an unbounded domain with a free surface and a fixed bottom. These data start with a flat surface and small velocity yet generate solutions in which the maximum vorticity gradient grows at a double-exponential rate while the solution exists. The construction accounts for gravity and surface tension and extends earlier results on vorticity stretching to the free-surface case and to unbounded domains. A reader would care because the result shows how small scales can form rapidly even from simple-looking initial states, bearing on questions of wave instability and the onset of complex fluid behavior.

What carries the argument

A specially chosen initial vorticity distribution that interacts with the free-surface boundary conditions to drive rapid stretching of vorticity gradients.

What would settle it

A calculation or simulation for the constructed initial data that shows the vorticity gradient remains bounded or grows only singly exponentially or slower would disprove the growth claim.

Watch

Extended reading notes

Core claim

We construct initial data with a flat free surface and small velocity such that the L^∞ norm of the vorticity gradient has at least a double-exponential growth rate within the lifespan of the corresponding solution. This work generalizes the result of Zlatos to the free-surface setting and Hu--Luo--Yao to the case of an unbounded domain.

Load-bearing premise

The chosen initial vorticity must allow the solution to exist long enough for the double-exponential growth to be realized.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript constructs explicit initial data with a flat free surface and small velocity for the 2D incompressible Euler equations with gravity and surface tension in an unbounded domain with fixed bottom. For this data the authors prove that the L^∞ norm of the vorticity gradient grows at least double-exponentially within the lifespan of the corresponding solution. The result generalizes Zlatos (fixed domain) to the free-surface setting and Hu-Luo-Yao (bounded domain) to the unbounded case.

Significance. If the central construction and growth estimate hold, the paper supplies a rigorous, parameter-free example of rapid small-scale creation in a physically relevant gravity-capillary free-surface model. The explicit initial-data choice and derivation from the Euler equations with kinematic/dynamic boundary conditions are strengths that allow direct verification of the double-exponential rate. This contributes concrete evidence toward understanding possible singularity formation in water-wave systems.

major comments (2)
  1. [§4] §4 (a-priori estimates for the free surface): The argument that the maximal existence time T* is at least as large as the time scale on which ||∇ω||_∞ reaches double-exponential size closes the Sobolev bounds on η only after invoking the growth of vorticity. Because the velocity is recovered from ω via a nonlocal operator whose kernel depends on the evolving surface height η, the constants in these estimates may deteriorate with the growth parameter; no explicit lower bound on T* independent of that parameter is displayed.
  2. [Theorem 1.1, §3] Theorem 1.1 and the construction in §3: The initial vorticity is chosen so that its transport produces the claimed growth, yet the proof that the corresponding solution remains regular up to the double-exponential time relies on the same a-priori estimates whose uniformity is questioned above. If those estimates fail to close, the observed growth may occur only after the solution has already broken down.
minor comments (2)
  1. [Introduction / Theorem 1.1] The statement of the main theorem would be clearer if the precise form of the initial vorticity (support, amplitude, and distance to the free surface) were displayed explicitly rather than described only qualitatively.
  2. [§2] Notation for the capillary term and the nonlocal operator recovering velocity from vorticity should be cross-referenced to the precise boundary conditions used in the estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the two major comments point by point below, providing clarifications on the uniformity of the estimates and indicating the revisions we will make.

read point-by-point responses

  1. Referee: [§4] §4 (a-priori estimates for the free surface): The argument that the maximal existence time T* is at least as large as the time scale on which ||∇ω||_∞ reaches double-exponential size closes the Sobolev bounds on η only after invoking the growth of vorticity. Because the velocity is recovered from ω via a nonlocal operator whose kernel depends on the evolving surface height η, the constants in these estimates may deteriorate with the growth parameter; no explicit lower bound on T* independent of that parameter is displayed.

    Authors: We appreciate this observation. The construction employs a small parameter ε > 0 that controls both the initial velocity size and the vorticity support. The double-exponential growth of ||∇ω||_∞ is realized on the time scale T ≈ log log(1/ε). On this interval the free-surface displacement η remains O(ε) because the velocity is recovered from the small initial vorticity via the nonlocal operator, whose dependence on η is controlled by the smallness of ε. Consequently, all Sobolev norms of η stay bounded by constants depending only on ε (and the fixed depth), independent of the growth in ∇ω. We will revise §4 to display an explicit lower bound T* ≥ c log log(1/ε) with c independent of ε (for ε sufficiently small), thereby closing the estimates uniformly and removing any circularity. revision: partial

  2. Referee: [Theorem 1.1, §3] Theorem 1.1 and the construction in §3: The initial vorticity is chosen so that its transport produces the claimed growth, yet the proof that the corresponding solution remains regular up to the double-exponential time relies on the same a-priori estimates whose uniformity is questioned above. If those estimates fail to close, the observed growth may occur only after the solution has already broken down.

    Authors: The initial data in §3 is constructed so that the vorticity is transported along trajectories that remain close to those of the fixed-domain problem of Zlatos, with free-surface perturbations controlled by the smallness of ε. The a-priori estimates of §4 are closed first by choosing ε small enough that the surface norms remain O(ε) on the interval [0, c log log(1/ε)]; only after these uniform bounds are secured do we invoke the transport of vorticity to obtain the double-exponential growth. Thus the solution stays regular at least up to the time when the growth is observed. We will add a short paragraph in the proof of Theorem 1.1 making this ordering of choices explicit. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; explicit construction yields growth via estimates

full rationale

The paper constructs explicit initial data (flat free surface, small velocity) and derives double-exponential growth of ||∇ω||_∞ directly from the 2D incompressible Euler equations coupled to gravity-capillary boundary conditions. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the growth follows from vorticity transport estimates and nonlocal velocity recovery that are independent of the target growth rate. Generalizations of Zlatos and Hu-Luo-Yao are to independent prior results with no author overlap or ansatz smuggling. The derivation is self-contained against the stated equations and does not rename known patterns or import uniqueness theorems from the same authors.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper relies on standard assumptions from ideal fluid dynamics; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

assumptions (2)
  • standard math The fluid is incompressible and inviscid, satisfying the 2D Euler equations
    Core modeling assumption for the water wave system.
  • domain assumption The free surface satisfies kinematic and dynamic boundary conditions incorporating gravity and surface tension
    Defines the gravity-capillary water wave model as stated.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Small scale creation in 2D gravity-capillary water waves with vorticity." pith.science (2026). https://pith.science/paper/2604.19380

@misc{pith2026260419380,
  author       = {Pith},
  title        = {Pith review of: Small scale creation in 2D gravity-capillary water waves with vorticity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.19380}},
  note         = {Machine review of arXiv:2604.19380}
}
abstract

In this paper, we consider 2D incompressible Euler equations in an unbounded domain with a free surface and a fixed bottom at finite depth. The fluid motion is under the influence of gravity and surface tension. We construct initial data with a flat free surface and small velocity, such that the $L^\infty$ norm of the vorticity gradient has at least a double-exponential growth rate within the lifespan of the corresponding solution. This work generalizes the result of Zlatos to the free-surface setting and Hu--Luo--Yao to the case of an unbounded domain.

Figures

Figures reproduced from arXiv: 2604.19380 by the authors.

Figure 1
Figure 1. Illustrations of our initial data ω0, D0, together with their evolution at later times. The four panels correspond to t = 0, 1, 2, 3 (from left to right and top to bottom). Here the blue and red colors represent positive and negative vorticity respectively; the black and red curves denote the free surface and the fixed bottom. As we show in Proposition 2.2, the free boundary Γt remains close to Γ0 throughout the lif… view at source ↗
Figure 2
Figure 2. The first figure illustrates an example of the truncated domain [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    On the water-wave equations with surface tension.Duke Math

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

    work page 2011

  2. [2]

    On the Cauchy problem for gravity water waves.Invent

    Thomas Alazard, Nicolas Burq, Claude Zuily. On the Cauchy problem for gravity water waves.Invent. Math., 198(1):71–163, 2014

    work page 2014

  3. [3]

    Global solutions and asymptotic behavior for two- dimensional gravity water waves.Ann

    Thomas Alazard, Jean-Marc Delort. Global solutions and asymptotic behavior for two- dimensional gravity water waves.Ann. Sci. ´Ec. Norm. Sup´ er.(4), 48(5):1149–1238, 2015

    work page 2015

  4. [4]

    Sulla propriet` a isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita.Atti Accad

    Ennio De Giorgi. Sulla propriet` a isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita.Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8), 5:33–44, 1958

    work page 1958

  5. [5]

    Well-posedness of the free-surface incompressible Euler equa- tions with or without surface tension.J

    Daniel Coutand, Steve Shkoller. Well-posedness of the free-surface incompressible Euler equa- tions with or without surface tension.J. Amer. Math. Soc., 20(3):829–930, 2007

    work page 2007

  6. [6]

    Ionescu, Benoˆ ıt Pausader, Fabio Pusateri

    Yu Deng, Alexandru D. Ionescu, Benoˆ ıt Pausader, Fabio Pusateri. Global solutions of the gravity-capillary water-wave system in three dimensions.Acta Math., 219(2):213–402, 2017

    work page 2017

  7. [7]

    Sergey Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation.Discrete Contin. Dyn. Syst., 23 (2009), 755–764

    work page 2009

  8. [8]

    Global solutions for the gravity water waves equation in dimension 3.Ann

    Pierre Germain, Nader Masmoudi, Jalal Shatah. Global solutions for the gravity water waves equation in dimension 3.Ann. Math., 175(2), 691–754, 2012

    work page 2012

Show all 34 references
  1. [9]

    Global existence for capillary water waves

    Pierre Germain, Nader Masmoudi, Jalal Shatah. Global existence for capillary water waves. Commun. Pure. Appl. Math., 68(4):625–687, 2015

    2015

  2. [10]

    Small scale creation for 2D free boundary Euler equations with surface tension.Ann

    Zhongtian Hu, Chenyun Luo, and Yao Yao. Small scale creation for 2D free boundary Euler equations with surface tension.Ann. PDE, 10(2), 2024. 19

    2024

  3. [11]

    Two-dimensional water waves in holomorphic coordinates II: global solutions.Bull

    Mihaela Ifrim, Daniel Tataru. Two-dimensional water waves in holomorphic coordinates II: global solutions.Bull. Soc. Math. France, 144(2):366–394, 2016

    2016

  4. [12]

    The lifespan of small data solutions in two-dimensional capillary water waves.Arch

    Mihaela Ifrim, Daniel Tataru. The lifespan of small data solutions in two-dimensional capillary water waves.Arch. Rational Mech. Anal., 225(3):1279–1346, 2017

    2017

  5. [13]

    Ionescu, Fabio Pusateri

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

    2015

  6. [14]

    Ionescu, Fabio Pusateri

    Alexandru D. Ionescu, Fabio Pusateri. Global regularity for 2D water waves with surface tension.Mem. Amer. Math. Soc., 256(1227), 2018

    2018

  7. [15]

    Small scale creation for solutions of the incompressible two-dimensional Euler equation.Ann

    Alexander Kiselev, Vladimir ˇSver´ ak. Small scale creation for solutions of the incompressible two-dimensional Euler equation.Ann. Math., 180(3), 1205–1220, 2014

    2014

  8. [16]

    Superlinear gradient growth for 2D Euler equation without boundary

    In-Jee Jeong, Yao Yao, Tao Zhou. Superlinear gradient growth for 2D Euler equation without boundary. arXiv:2507.15739, preprint, 2025

    2025

  9. [17]

    Well-posedness of the water-waves equations.J

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

    2005

  10. [18]

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

    Hans Lindblad. Well-posedness for the motion of an incompressible liquid with free surface boundary.Ann. Math., 162(1), 109–194, 2005

    2005

  11. [19]

    Well-posedness of the water-wave problem with surface tension.J

    Mei Ming, Zhifei Zhang. Well-posedness of the water-wave problem with surface tension.J. Math. Pures Appl., 92(5), 429–455, 2009

    2009

  12. [20]

    Geometry and a priori estimates for free boundary problems of the Euler’s equation.Commun

    Jalal Shatah, Chongchun Zeng. Geometry and a priori estimates for free boundary problems of the Euler’s equation.Commun. Pure. Appl. Math., 61(5), 698–744, 2008

    2008

  13. [21]

    A priori estimates for fluid interface problems.Commun

    Jalal Shatah, Chongchun Zeng. A priori estimates for fluid interface problems.Commun. Pure. Appl. Math., 61(6), 848–876, 2008

    2008

  14. [22]

    Local well-posedness for fluid interface problems.Arch

    Jalal Shatah, Chongchun Zeng. Local well-posedness for fluid interface problems.Arch. Rational Mech. Anal., 199(2), 653–705, 2011

    2011

  15. [23]

    Long time behavior of 2D water waves with point vortices.Commun

    Qingtang Su. Long time behavior of 2D water waves with point vortices.Commun. Math. Phys., 380(3):1173–1266, 2020

    2020

  16. [24]

    Global infinite energy solutions for the 2D gravity water waves system

    Xuecheng Wang. Global infinite energy solutions for the 2D gravity water waves system. Commun. Pure. Appl. Math., 71(1):90–162, 2018

    2018

  17. [25]

    Global regularity for the 3D finite depth capillary water waves.Ann

    Xuecheng Wang. Global regularity for the 3D finite depth capillary water waves.Ann. Sci. ´Ec. Norm. Sup´ er., 53(4):847–893, 2020

    2020

  18. [26]

    Well-posedness in Sobolev spaces of the full water wave problem in 2-D.Invent

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

    1997

  19. [27]

    Well-posedness in Sobolev spaces of the full water wave problem in 3-D.J

    Sijue Wu. Well-posedness in Sobolev spaces of the full water wave problem in 3-D.J. Amer. Math. Soc., 12(2):445–495, 1999

    1999

  20. [28]

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

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

    2009

  21. [29]

    Global wellposedness of the 3-D full water wave problem.Invent

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

    2011

  22. [30]

    Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow.J

    Xiaoqian Xu. Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow.J. Math. Anal. Appl., 439(2), 594–607, 2016

    2016

  23. [31]

    On the free boundary problem of three-dimensional incompressible Euler equations.Commun

    Ping Zhang, Zhifei Zhang. On the free boundary problem of three-dimensional incompressible Euler equations.Commun. Pure. Appl. Math., 61(7):877–940, 2008. 20

    2008

  24. [32]

    Exponential growth of the vorticity gradient for the Euler equation on the torus.Adv

    Andrej Zlatoˇ s. Exponential growth of the vorticity gradient for the Euler equation on the torus.Adv. Math., 268:396–403, 2015

    2015

  25. [33]

    Maximal double-exponential growth for the Euler equation on the half-plane

    Andrej Zlatoˇ s. Maximal double-exponential growth for the Euler equation on the half-plane. Invent. Math., 243(1):117–126, 2026

    2026

  26. [34]

    Long-term regularity of 3D gravity water waves.Commun

    Fan Zheng. Long-term regularity of 3D gravity water waves.Commun. Pure Appl. Math., 75(5):1074–1180, 2022. 21

    2022

Pith tools

Reviewed May 10, 2026 · model on record in the stance chip above.