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arxiv: 2301.04599 · v4 · submitted 2023-01-11 · 🧮 math.AP

Uniform in gravity estimates for 2D water waves

Siddhant Agrawal This is my paper

Pith reviewed 2026-05-24 09:23 UTC · model grok-4.3

classification 🧮 math.AP
keywords 2D water waveslocal wellposednesssingular interfacesscaling invariant estimatesgravity water wavesblow-up criterionangled crestsfree boundary problems
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The pith

Local wellposedness holds for 2D water waves with corners and cusps uniformly as gravity vanishes.

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

The paper proves local wellposedness for the two-dimensional gravity water waves equation on infinite domains when the initial interface contains corners or cusps. The time of existence remains positive and independent of the gravity parameter even as it approaches zero. A scaling-invariant energy estimate supplies the necessary control and also yields an improved blow-up criterion plus existence at zero gravity. The same estimate applied to the gravity-free equation on domains homeomorphic to a disc permits angled crests and cusps yet shows that the energy of some such data must blow up in finite time, establishing sharpness.

Core claim

We prove a local wellposedness result which allows interfaces with corners and cusps as initial data and which is such that the time of existence of solutions is uniform even as the gravity parameter g → 0. Moreover the energy estimate used to prove this result is scaling invariant. For g>0 we prove an improved blow up criterion and an existence result for g=0. As an application we consider the water wave equation with no gravity where the fluid domain is homeomorphic to the disc; we prove local wellposedness allowing angled crests and cusps and show by a rigidity argument that there exist initial interfaces with angled crests for which the energy blows up in finite time.

What carries the argument

Scaling-invariant energy estimate that controls solutions with singular interfaces uniformly in the gravity parameter.

If this is right

  • Existence time stays bounded away from zero independently of how small g becomes.
  • Improved blow-up criteria hold for positive gravity with singular initial data.
  • Local wellposedness holds for the gravity-free equation on disc-like domains, allowing angled crests and cusps.
  • Certain angled-crest initial data cause finite-time energy blow-up, showing the result is sharp.
  • For smooth data with small initial velocity the existence time is longer than in earlier work.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The uniformity may let one pass directly from positive-gravity solutions to the zero-gravity limit without losing existence-time control.
  • The rigidity argument for blow-up could adapt to other free-boundary problems that possess similar scaling.
  • Numerical schemes for water waves might retain sharp features without gravity-dependent regularization.
  • The scaling invariance could support uniform estimates when additional parameters such as surface tension are varied.

Load-bearing premise

The energy estimate stays scaling invariant when the interface contains corners or cusps.

What would settle it

A family of solutions starting from cornered interfaces whose maximal existence time shrinks to zero as g approaches zero would disprove the uniformity.

Figures

Figures reproduced from arXiv: 2301.04599 by Siddhant Agrawal.

Figure 1
Figure 1. Figure 1: An example of blow up of the energy Ee(t) Theorem 1.2. (Informal) Consider an initial data with Ee(0) < ∞. Then there exists a time T & 1 Ee(0) 1 2 such that there exists a unique solution to (2) in [0, T]. If T ∗ > 0 is the maximal time of existence, then either T ∗ = ∞ or lim supt→T ∗ Be(t) = ∞. Moreover there exist initial data with Ee(0) < ∞ for which the maximal time of existence T ∗ is finite with T … view at source ↗
read the original abstract

We consider the 2D gravity water waves equation on an infinite domain. We prove a local wellposedness result which allows interfaces with corners and cusps as initial data and which is such that the time of existence of solutions is uniform even as the gravity parameter $g \to 0$. For $g>0$, we prove an improved blow up criterion for these singular solutions and we also prove an existence result for $g = 0$. Moreover the energy estimate used to prove this result is scaling invariant. As an application of this energy estimate, we then consider the water wave equation with no gravity where the fluid domain is homeomorphic to the disc. We prove a local wellposedness result which allows for interfaces with angled crests and cusps as initial data and then by a rigidity argument, we show that there exists initial interfaces with angled crests for which the energy blows up in finite time, thereby proving the optimality of this local wellposedness result. For smooth initial data, this local wellposedness result gives a longer time of existence as compared to previous results when the initial velocity is small and we also improve upon the blow up criterion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves local well-posedness for the 2D gravity water-wave system on the infinite strip, allowing initial interfaces with corners and cusps; the existence time is uniform down to g=0. The proof rests on a scaling-invariant a-priori energy estimate. For g>0 an improved blow-up criterion is obtained; existence is also shown for g=0. As an application, the authors treat the gravity-free problem on a disc-homeomorphic domain, obtain local well-posedness for angled crests and cusps, and then use a rigidity argument to exhibit initial data with angled crests whose energy blows up in finite time, proving optimality of the result. For smooth data the existence time is longer than previous results when the initial velocity is small, and the blow-up criterion is sharpened.

Significance. If the scaling-invariant estimate closes for the indicated singular data, the result would be a substantial advance: it supplies the first uniform-in-g local theory that tolerates corners and cusps, gives an optimality statement via explicit blow-up, and improves the smooth-data lifespan. The scaling invariance itself, if parameter-free and free of angle-dependent constants, is a technically noteworthy feature.

major comments (2)
  1. [energy estimate section] The central claim of g-uniform existence for corner/cusp data rests on the scaling-invariant energy estimate. The manuscript must verify that the commutator and multiplier estimates used to close this estimate remain uniform when the interface has a corner or cusp; any implicit C^{1,α} assumption away from the singularity or any weight that degenerates at the crest would make the constant depend on the opening angle or on g, destroying uniformity. (See the statement of the main a-priori estimate and the function-space definitions in the section containing the energy estimate.)
  2. [application to disc-homeomorphic domain] The rigidity argument that produces finite-time energy blow-up for certain angled-crest data must be checked for compatibility with the function spaces in which local well-posedness is proved; if the blow-up example lies outside the space for which the a-priori estimate is derived, it does not establish optimality of the local theory. (See the paragraph containing the rigidity construction and the comparison with the local-well-posedness theorem.)
minor comments (2)
  1. [introduction] The abstract states that the energy estimate is scaling invariant but does not record the precise function spaces or the precise scaling; this information should appear already in the introduction so that the reader can immediately assess the claim.
  2. Notation for the singular points (corners versus cusps) and for the opening angle should be fixed once and used consistently throughout the statements of the theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we respond point-by-point to the two major comments, clarifying the uniformity of the estimates and the compatibility of the blow-up construction with the function spaces used for local well-posedness.

read point-by-point responses

  1. Referee: [energy estimate section] The central claim of g-uniform existence for corner/cusp data rests on the scaling-invariant energy estimate. The manuscript must verify that the commutator and multiplier estimates used to close this estimate remain uniform when the interface has a corner or cusp; any implicit C^{1,α} assumption away from the singularity or any weight that degenerates at the crest would make the constant depend on the opening angle or on g, destroying uniformity. (See the statement of the main a-priori estimate and the function-space definitions in the section containing the energy estimate.)

    Authors: The commutator and multiplier estimates in the energy section are obtained via a paradifferential calculus adapted to the corner/cusp geometry; the symbols and weights are chosen so that no C^{1,α} regularity is assumed away from the crest and the constants remain independent of both the opening angle and g. The scaling-invariant structure of the a-priori estimate is used precisely to guarantee this uniformity, and the function-space definitions incorporate this feature explicitly. We will add a short clarifying paragraph summarizing these uniformity checks. revision: partial

  2. Referee: [application to disc-homeomorphic domain] The rigidity argument that produces finite-time energy blow-up for certain angled-crest data must be checked for compatibility with the function spaces in which local well-posedness is proved; if the blow-up example lies outside the space for which the a-priori estimate is derived, it does not establish optimality of the local theory. (See the paragraph containing the rigidity construction and the comparison with the local-well-posedness theorem.)

    Authors: The rigidity construction is performed with initial data that satisfy exactly the same Sobolev regularity, compatibility conditions, and weighted estimates required by the local well-posedness theorem on the disc-homeomorphic domain. Consequently the finite-time blow-up occurs inside the function space for which the a-priori estimate is derived, establishing optimality of the local theory. revision: no

Circularity Check

0 steps flagged

No circularity; scaling-invariant estimate presented as independent tool

full rationale

The abstract and described claims establish local well-posedness via a scaling-invariant energy estimate that is invoked to obtain g-uniform existence times for singular data. This estimate is not shown to be self-definitional, a renamed fit, or dependent on a load-bearing self-citation chain. The derivation chain remains self-contained against external benchmarks, with the invariance serving as an independent technical input rather than a redefinition of the target well-posedness result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions for incompressible irrotational flow plus the scaling invariance of a specific energy functional; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The fluid is incompressible and irrotational on a domain whose boundary is a graph or closed curve with possible corners/cusps.
    This is the standard setup invoked for the 2D gravity water waves equation.
  • ad hoc to paper The energy estimate employed is scaling invariant.
    Explicitly highlighted in the abstract as the property that yields uniformity as g→0.

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