Pith. sign in

REVIEW 3 major objections 3 minor 1 cited by

Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Finite-time blow-up in the free-boundary Euler equations with surface tension is classified into five concrete mechanisms.

desk verdict New classification theorem, but a false interpolation inequality (3.23) breaks the energy closure; needs major revision. read the letter →

arxiv 2507.10032 v2 pith:AJLFU6NV submitted 2025-07-14 math.AP

Chengchun Hao , Tao Luo , Siqi Yang This is my paper · ORCID
classification math.AP MSC 35Q3535R3535B4476B0376B45
keywords freeboundaryproblemincompressibleEulerequationssurfacetensionfinite-timeblow-upcriterionsplashsingularityself-intersectionvorticity
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 proves that a strong solution of the three-dimensional incompressible Euler equations with surface tension, in a bounded domain with a closed free boundary, cannot reach a finite maximal existence time unless at least one of five concrete mechanisms occurs: the boundary first self-intersects, mean curvature or boundary regularity is lost, the normal boundary velocity loses smoothness, or the velocity gradient accumulates in an integrable way on the boundary or inside the fluid. This matters because earlier blow-up criteria required graphs, symmetries, or periodicity, which cannot describe generic singularities such as folding or pinch-off; the present criterion imposes none of these assumptions. The proof runs by contradiction: if all five mechanisms are absent, a uniform interior and exterior ball radius and elliptic regularity estimates keep an energy functional uniformly bounded, so the solution extends past the supposed blow-up time. For simply connected domains the interior condition is refined to a vorticity condition, and for irrotational flows singularity can occur only at the boundary.

What carries the argument

The argument is carried by the energy functional $E(t)=\frac12\left(\int_{\Omega_t}|D_t^2 v|^2\,dx+\int_{\partial\Omega_t}|\bar{\nabla}(D_t v\cdot n)|^2\,dS+\int_{\Omega_t}|\nabla^2(\nabla\times v)|^2\,dx\right)$, together with an equivalent functional containing $\|D_t v\|_{H^{3/2}(\Omega_t)}$ and $\|v\|_{H^3(\Omega_t)}$. Surface tension enters through the boundary condition $p=H_{\partial\Omega_t}$, so the material derivative $D_t v$ behaves like a $3/2$-order spatial derivative, and tracking $D_t^2 v$ together with the boundary term $\bar{\nabla}(D_t v\cdot n)$ produces the differential inequality $\frac{d}{dt}E(t)\le C(\|\nabla v\|_{L^\infty(\Omega_t)}+\|\nabla v\|_{L^\infty(\partial\Omega_t)}+1)E(t)$. If no blow-up scenario occurs, the uniform ball radius condition $\inf_{0\le t<T^*}R(\Omega_t)>C^{-1}$ and the elliptic estimates of Lemmas 2.6-2.8 keep the constants uniform, the standard exponential-integral bound gives a uniform estimate on $E$ up to $T^*$, and Lemma 2.7 lifts the boundary regularity so the solution can be extended, a contradiction.

What would settle it

Construct or numerically observe a family of solutions approaching a finite time $T^*$ at which all five quantities stay bounded (boundary remains embedded with uniform ball radius, mean curvature in $H^{3/2}$, normal velocity in $H^{5/2}$, and both $L^1_tL^\infty$ velocity-gradient integrals are finite) yet the $H^3\times H^4$ norms fail to converge; that would contradict Theorem 1.1. A more targeted check is whether $\inf_{0\le t<T^*}R(\Omega_t)$ can vanish while none of the five alternatives occurs.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for a strong solution with $v\in C([0,T^*);H^3(\Omega_t))$ and $\partial\Omega_t\in C([0,T^*);H^4)$, if the maximal existence time $T^*$ is finite then at least one of five alternatives holds: (1) first self-intersection of the free boundary at $t=T^*$; (2) loss of mean curvature regularity in $H^{3/2}$ or of boundary regularity in $H^{2+\varepsilon}$ for any sufficiently small fixed $\varepsilon>0$; (3) loss of $H^{5/2}$ regularity of the normal boundary velocity; (4) $\int_0^{T^*}\|\nabla v\|_{L^\infty(\partial\Omega_t)}\,dt=\infty$; or (5) $\int_0^{T^*}\|\nabla v\|_{L^\infty(\Omega_t)}\,dt=\infty$. The proof shows these are the only ways a singularity can form, with no symmetry, graph, periodicity, or simple-connectivity assumptions. For simply connected domains, alternative (5) is refined to the vorticity condition $\limsup_{t\to T^*}\|\nabla\times v\|_{L^2(\Omega_t)}+\int_0^{T^*}\|\nabla\times v\|_{L^\infty(\Omega_t)}\,dt=\infty$; for irrotational flows only the boundary alternatives remain.

Load-bearing premise

The argument depends on the free boundary maintaining a uniformly rounded shape, meaning a fixed lower bound on the interior and exterior ball radius, and on the elliptic regularity constants staying uniform in time under that geometric control; if these degenerate before any of the five listed mechanisms occurs, the energy estimate no longer controls the solution.

Editorial extensions

If this is right

  • All finite-time singularities of these surface-tension free-boundary Euler solutions are captured by five independent mechanisms, so no additional hidden singularity scenario is needed.
  • Blow-up criteria can be stated without graph, symmetry, or topology assumptions, so turning, folding, multiply connected domains, and non-graph boundaries are included.
  • In simply connected domains, interior blow-up is governed entirely by vorticity, and irrotational flows can break only at the free boundary.
  • In the fixed-boundary case the criterion reduces to a classical vorticity accumulation condition for the velocity gradient.
  • The gap between $H^{3/2}$ curvature regularity and $H^{2+\varepsilon}$ boundary regularity is intrinsic to recovering boundary regularity from mean curvature through elliptic estimates.

Reading between the lines

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

  • Editorial: the five quantities suggest a concrete numerical diagnostic for imminent singularity formation, and in simply connected geometries the interior integral can be replaced by a vorticity monitor.
  • Editorial: the uniform ball radius is the geometric linchpin; a boundary that degenerates into a cusp or near-contact without self-intersection would force one of the regularity-loss alternatives, so the classification implicitly predicts how curvature and injectivity failures must be coupled.
  • Editorial: the energy and commutator structure is not tied to the specific pressure law, so a similar five-mechanism classification may hold for other surface-tension free-boundary systems such as charged liquid drops or ideal MHD with surface tension.
  • Editorial: the separation of tangential boundary gradient blow-up from interior gradient blow-up suggests that boundary-layer-type singularity can occur while the interior remains smooth, a distinction that could be tested in numerical simulations.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper claims a complete classification of finite-time blow-up scenarios for the 3D incompressible free-boundary Euler equations with surface tension, at the H^3 × H^4 regularity level, in bounded domains with closed boundary, without symmetry, periodicity, graph, or simple-connectivity assumptions. Theorem 1.1 asserts that if the maximal existence time T* is finite, at least one of five scenarios must occur: first self-intersection of the free boundary; loss of mean-curvature regularity in H^{3/2} or boundary regularity in H^{2+ε}; loss of H^{5/2} regularity of the normal boundary velocity; L^1_t L^∞ blow-up of the tangential velocity gradient on the boundary; or L^1_t L^∞ blow-up of the interior velocity gradient. Theorem 1.5 refines the interior alternative, under simple connectivity, to a vorticity-based criterion. The proof is by contradiction via a high-order energy functional E(t) involving D_t^2 v, the tangential gradient of D_t v·n, and ∇^2(∇×v), and aims to close a Grönwall inequality under the assumption that all five scenarios are absent.

Significance. If the main theorem were established, it would be a significant advance in the theory of free-boundary Euler equations with surface tension: prior blow-up criteria in this setting either required graph representations, higher (H^6) regularity, or stronger pointwise-in-time control of the velocity gradient. The paper's dynamic-reference-surface methodology and the proposed separation of boundary and interior singularity mechanisms are conceptually interesting, and the authors are to be credited for attempting a general geometric framework. However, the central energy estimate rests on a false interpolation inequality, so Theorem 1.1 is not proved. In addition, the abstract overstates the fixed-boundary BKM recovery: the paper's own equation (1.9) includes an enstrophy term beyond the classical BKM criterion (1.10). These are load-bearing issues, and the manuscript cannot be accepted in its present form.

major comments (3)
  1. [Section 3, Eq. (3.23)] The inequality ∥v∥²_{W^{3/2,4}} ≤ C∥v∥_{L^6}∥v∥_{H^3} asserted in (3.23) is not a valid Gagliardo–Nirenberg interpolation. Solving 1/4 = θ/6 + (1−θ)/2 forces θ = 3/4, and then the interpolation condition s = θ·0 + (1−θ)·3 gives s = 3/4, not 3/2. A scaling counterexample confirms the failure: for v_λ(x) = λ^{-1/2}φ(x/λ), the left-hand side is of order λ^{-5/2} while the right-hand side is of order λ^{-2}. This estimate is used in Lemma 3.10 to control the cubic term ∂_i v_j ∂_j v_k ∂_k v_i in H^{1/2}; the correct available bound introduces an extra factor ∥∇v∥_{L∞}. Consequently, the I_2 estimate in Proposition 3.11 acquires a superlinear term, and the Grönwall inequality (3.25) does not close. The contradiction argument for Theorem 1.1 therefore collapses.
  2. [Abstract and Section 1.1 (Eq. (1.9))] The abstract states that the simply-connected refinement 'recovers exactly the classical Beale–Kato–Majda criterion in the fixed-boundary case,' but Section 1.1 and equation (1.9) explicitly state that the fixed-boundary criterion obtained here (limsup_{t→T*}∥∇×v∥_{L²} + ∫_0^{T*}∥∇×v∥_{L∞}dt = ∞) differs from the classical BKM criterion (1.10) by incorporating enstrophy dynamics. The abstract's claim of exact recovery is therefore inaccurate and should be corrected to match the actual theorem.
  3. [Section 4, proof of Theorem 1.5] The proof of Theorem 1.5 inherits the failure of the energy closure from Proposition 3.11: the step from (4.4) to (4.6) relies on the Grönwall inequality (3.33), which depends on the faulty estimate (3.23). Even if the interpolation error were repaired in a way that still yields a linear Grönwall inequality, the additional factor ∥∇v∥_{L∞} would prevent the closure, so the vorticity-based refinement is not established by the arguments presented.
minor comments (3)
  1. [Abstract vs Theorem 1.1] The abstract lists four mechanisms (i)–(iv), whereas Theorem 1.1 lists five scenarios; scenario (4), the L^1_t L^∞ blow-up of the tangential velocity gradient on the boundary, is omitted in the abstract. The numbering should be harmonized.
  2. [Section 3, Eq. (3.8)] In the display after (3.8), the statement 'the unit outer normal vector n ∈ H^{5/2}' should specify the trace space H^{5/2}(∂Ω_t) rather than the domain space, to avoid ambiguity.
  3. [Section 3, Lemma 3.9] In the estimate for the first term of (3.17), the notation ∥D_t^2 v·n∥_{H^{-1/2}(∂Ω_t)} is used; it would be clearer to state explicitly that this is the dual norm with respect to the H^{1/2}(∂Ω_t) pairing, even though the referenced normal trace theorem justifies the bound.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem 1.1 is established by a standard contradiction-and-continuation argument; the only self-citation ([31]) is contextual and not load-bearing.

full rationale

The derivation chain is self-contained against external benchmarks. The proof of Theorem 1.1 assumes the negation of the five scenarios, which yields uniform bounds (3.1)-(3.4), then derives energy estimates (3.25), (3.30) using cited elliptic/div-curl lemmas (Lemmas 2.6-2.9 from [38], [47], [14]) and Grönwall's inequality, and finally extends the solution past T*. The classification is not equivalent to its inputs by construction: the five alternatives are exhaustive because their simultaneous failure supplies the a priori control needed for continuation, which is exactly the logical form of a blow-up criterion. No parameter is fitted and no fitted quantity is renamed as a prediction. The only self-citation is [31] (Hao-Yang), used in the introduction and Remark 3.6 to attribute the dynamically-updated reference-surface methodology and the 3/2-order material-derivative scaling; the actual energy estimates in Section 3 are derived in the paper and rest on [38], [47], and [14], not on [31]. Per the rubric this one minor non-load-bearing self-citation gives score 2 rather than 0. The reviewer-level concern that (3.23) misapplies Gagliardo-Nirenberg interpolation, and the concern that the uniform ball-radius assumption (3.1) may degenerate without the listed mechanisms, are correctness/rigor risks, not circularity: even if (3.23) is false, the claimed conclusion does not become identical to the hypotheses by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The constant C(C†) in the estimates depends only on the a priori bound C† and on the initial data; it is not an adjustable parameter. The axioms are prior theorems and standard elliptic/geometric estimates, plus the geometric control assumption that the boundary does not degenerate between isolated mechanisms. No new entities are postulated.

assumptions (4)
  • domain assumption Local well-posedness for the free-boundary Euler equations with surface tension at H^3 × H^4 regularity holds on a maximal time interval [0,T*).
    Invoked in Theorem 1.1's hypotheses and used to justify continuation once a priori bounds are obtained. Cited to Shatah-Zeng [49].
  • standard math The elliptic regularity and div-curl estimates of Lemmas 2.6-2.9 hold under the stated H^{2+ε} boundary and H^{3/2} curvature hypotheses.
    These lemmas are imported from [38, Prop 2.12, Thm 3.1, Prop 3.8], [47, Prop A.2], and [14, Thm 1.3]; the main proof does not reprove them.
  • standard math Ferrari's logarithmic estimate (Lemma 4.1, equation (4.2)) is valid for simply connected bounded domains with C^{2,α} boundary, with constant depending only on the ball-radius bound C†.
    Used in Theorem 1.5 to convert vorticity bounds into W^{1,∞} velocity bounds. Cited to Ferrari [26].
  • domain assumption The uniform ball-radius condition inf R(Ω_t) > C^{-1} follows from the negation of self-intersection together with uniform H^{2+ε} boundary regularity, and it ensures uniform C^{2,α} geometry for all t < T*.
    Section 3, equations (3.1) and the discussion after it; this geometric control underpins the elliptic estimates and the domain-dependent norms.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension." pith.science (2026). https://pith.science/paper/AJLFU6NV

@misc{pith2026250710032,
  author       = {Pith},
  title        = {Pith review of: Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AJLFU6NV}},
  note         = {Machine review of arXiv:2507.10032}
}
abstract

We establish a blow-up criterion for strong solutions of the three-dimensional incompressible Euler equations with surface tension in a bounded domain with a closed moving free boundary. The criterion is formulated at the $H^3\times H^4$ regularity level of the Shatah--Zeng local well-posedness theory and imposes \textit{no} assumptions of symmetry, periodicity, graph structure, or simple connectedness. If the maximal existence time $T<\infty$, then at least one of the following four mechanisms must occur: (i) first self-intersection of the free boundary; (ii) loss of mean curvature regularity in $H^{\frac{3}{2}}$, or loss of boundary regularity in $H^{2+\varepsilon}$ for any sufficiently small fixed $\varepsilon>0$; (iii) loss of $H^{\frac{5}{2}}$ regularity of the normal boundary velocity; or (iv) $L^1_tL^\infty$ blow-up of the interior velocity gradient. For simply connected domains, the interior alternative admits a refinement involving only the $L^1_tL^\infty$-norm of the vorticity, and this refinement recovers exactly the classical Beale--Kato--Majda criterion in the fixed-boundary case. For irrotational flows in the simply connected free-boundary setting, the criterion reduces to the three boundary mechanisms.

Figures

Figures reproduced from arXiv: 2507.10032 by the authors.

Figure 1
Figure 1. Bounded fluid domain with a closed free surface. 2020 Mathematics Subject Classification. 35Q35, 35R35, 35B44, 76B03, 76B45. Key words and phrases. free boundary problem, incompressible Euler equations, finite-time blow-up, splash singular￾ity, surface tension. 1 arXiv:2507.10032v1 [math.AP] 14 Jul 2025 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Some cases of self-intersection of the free boundary. The formation of these self-intersecting singularities typically preserves solution regularity, with blow-up criteria for the incompressible rotational Euler equations systematically established when surface tension is neglected: the graph-based framework for bounded domains [51], the result for initial domains diffeomorphic to a ball [29], and particularly the s… view at source ↗
Figure 3
Figure 3. Two scenarios under the graph assumption on the torus T 2 : on the left, bound￾ary turning occurs, and on the right, the upper free boundary contacts the bottom. To characterize self-intersection of the free boundary, the first and third authors introduced a dy￾namically updated reference surface methodology [31], which preserves non-degenerate coordinate map￾ping even as the interface approaches self-contact. This … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The regularity loss of the free boundary and its normal velocity. (i) Cases (1)–(3) become trivial: Geometric evolution quantities (self-intersection, mean curvature, normal velocity vn) are time-independent by assumption. (ii) Lemma 2.5 no longer requires tracking mea…
Figure 5
Figure 5. Figure 5: Self-intersection with bounded curvature variation. Conversely, in squeeze singularities ( [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Curvature blow-up accompanying the self-intersection of the free boundary [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: The free boundary approaches self-intersection without curvature blow-up. governs the condition in (1.1c). Consequently, such approaches either yield degenerate physi￾cal models (applying (1.1c) to curvature-erased surfaces), or generate analytically intractable system…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions

    math.AP 2025-06 conditional novelty 6.0 of 10

    For the 2D free-boundary Euler equations with surface tension on a droplet, there exist arbitrarily small initial velocities such that the vorticity Hessian grows at least exponentially in time.

Reference graph

Works this paper leans on

55 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [31]

    Hao, C., Yang, S.:A prioriestimates and a blow-up criterion for the incompressible ideal MHD equations with surfacetensionandaclosedfreesurface.Nonlinearity38(7)PaperNo.075009(2025).doi:10.1088/1361-6544/addd5a

    doi:10.1088/1361-6544/addd5a 2025

  2. [1]

    Duke Math

    Alazard, T., Burq, N., Zuily, C.: On the water-wave equations with surface tension. Duke Math. J.158(3) 413–499 (2011). doi:10.1215/00127094-1345653

    work page doi:10.1215/00127094-1345653 2011

  3. [2]

    Alazard, T., Burq, N., Zuily, C.: On the Cauchy problem for gravity water waves. Invent. Math.198(1) 71–163 (2014). doi:10.1007/s00222-014-0498-z

    work page doi:10.1007/s00222-014-0498-z 2014

  4. [3]

    Alazard, T., Delort, J.M.: Global solutions and asymptotic behavior for two dimensional gravity water waves. Ann. Sci. Éc. Norm. Supér. (4)48(5) 1149–1238 (2015). doi:10.24033/asens.2268

    work page doi:10.24033/asens.2268 2015

  5. [4]

    Ambrose, D.M., Masmoudi, N.: The zero surface tension limit of two-dimensional water waves. Comm. Pure Appl. Math.58(10) 1287–1315 (2005). doi:10.1002/cpa.20085

    work page doi:10.1002/cpa.20085 2005

  6. [5]

    Indiana Univ

    Ambrose, D.M., Masmoudi, N.: The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J.58(2) 479–521 (2009). doi:10.1512/iumj.2009.58.3450

    work page doi:10.1512/iumj.2009.58.3450 2009

  7. [6]

    Aydin, M.S., Kukavica, I., Ożański, W.S., Tuffaha, A.: Construction of the free-boundary 3D incompressible Euler flow under limited regularity. J. Differential Equations394209–236 (2024). doi:10.1016/j.jde.2024.02.027

    work page doi:10.1016/j.jde.2024.02.027 2024

  8. [7]

    Beale, J.T.: The initial value problem for the Navier-Stokes equations with a free surface. Comm. Pure Appl. Math. 34(3) 359–392 (1981). doi:10.1002/cpa.3160340305

    work page doi:10.1002/cpa.3160340305 1981

Show all 55 references
  1. [8]

    Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the3-D Euler equations. Comm. Math. Phys.94(1) 61–66 (1984). doi:10.1007/BF01212349

    1984 doi

  2. [9]

    Brezis, H., Mironescu, P.: Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Ann. Inst. H. Poincaré C Anal. Non Linéaire35(5) 1355–1376 (2018). doi:10.1016/j.anihpc.2017.11.007

    2018 doi

  3. [10]

    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Finite time singularities for water waves with surface tension. J. Math. Phys.53(11) 115622, 26 (2012). doi:10.1063/1.4765339

    2012 doi

  4. [11]

    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Finite time singularities for the free boundary incompressible Euler equations. Ann. of Math. (2)178(3) 1061–1134 (2013). doi:10.4007/annals.2013. 178.3.6

    2013 doi

  5. [12]

    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Splash singularities for the free boundary Navier-Stokes equations. Ann. PDE5(1) Paper No. 12, 117 (2019). doi:10.1007/s40818-019-0068-1

    2019 doi

  6. [13]

    Cheng, C.H.A., Coutand, D., Shkoller, S.: On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity. Comm. Pure Appl. Math.61(12) 1715–1752 (2008). doi:10.1002/cpa.20240

    2008 doi

  7. [14]

    Cheng, C.H.A., Shkoller, S.: Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains. J. Math. Fluid Mech.19(3) 375–422 (2017). doi: 10.1007/s00021-016-0289-y

    2017 doi

  8. [15]

    Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Comm. Pure Appl. Math.53(12) 1536–1602 (2000). doi:10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.3.CO;2-H

    2000 doi

  9. [16]

    Córdoba, A., Córdoba, D., Gancedo, F.: Interface evolution: water waves in 2-D. Adv. Math.223(1) 120–173 (2010). doi:10.1016/j.aim.2009.07.016 28 CHENGCHUN HAO, TAO LUO, AND SIQI YANG

    2010 doi

  10. [17]

    Córdoba, D., Enciso, A., Grubic, N.: Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations. Ann. PDE7(1) Paper No. 12, 40 (2021). doi:10.1007/s40818-021-00101-6

    2021 doi

  11. [18]

    Coutand, D.: Finite-timesingularityformationforincompressibleEulermovinginterfacesintheplane.Arch.Ration. Mech. Anal.232(1) 337–387 (2019). doi:10.1007/s00205-018-1322-5

    2019 doi

  12. [19]

    Coutand, D., Lindblad, H., Shkoller, S.: A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum. Comm. Math. Phys.296(2) 559–587 (2010). doi:10.1007/s00220-010-1028-5

    2010 doi

  13. [20]

    Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc.20(3) 829–930 (2007). doi:10.1090/S0894-0347-07-00556-5

    2007 doi

  14. [21]

    Coutand, D., Shkoller, S.: On the finite-time splash and splat singularities for the 3-D free-surface Euler equations. Comm. Math. Phys.325(1) 143–183 (2014). doi:10.1007/s00220-013-1855-2

    2014 doi

  15. [22]

    Coutand, D., Shkoller, S.: On the splash singularity for the free-surface of a Navier-Stokes fluid. Ann. Inst. H. Poincaré C Anal. Non Linéaire36(2) 475–503 (2019). doi:10.1016/j.anihpc.2018.06.004

    2019 doi

  16. [23]

    Acta Math.219(2) 213–402 (2017)

    Deng, Y., Ionescu, A.D., Pausader, B., Pusateri, F.: Global solutions of the gravity-capillary water-wave system in three dimensions. Acta Math.219(2) 213–402 (2017). doi:10.4310/ACTA.2017.v219.n2.a1

    2017 doi

  17. [24]

    Di Iorio, E., Marcati, P., Spirito, S.: Splash singularities for a general Oldroyd model with finite Weissenberg number. Arch. Ration. Mech. Anal.235(3) 1589–1660 (2020). doi:10.1007/s00205-019-01451-z

    2020 doi

  18. [25]

    Disconzi, M.M., Kukavica, I., Tuffaha, A.: A Lagrangian interior regularity result for the incompressible free bound- ary Euler equation with surface tension. SIAM J. Math. Anal.51(5) 3982–4022 (2019). doi:10.1137/18M1216808

    2019 doi

  19. [26]

    Ferrari, A.B.: On the blow-up of solutions of the3-D Euler equations in a bounded domain. Comm. Math. Phys. 155(2) 277–294 (1993)

    1993

  20. [27]

    Fusco, N., Julin, V., Morini, M.: The surface diffusion flow with elasticity in three dimensions. Arch. Ration. Mech. Anal.237(3) 1325–1382 (2020). doi:10.1007/s00205-020-01532-4

    2020 doi

  21. [28]

    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). doi:10.4007/annals.2012.175.2.6

    2012 doi

  22. [29]

    Ginsberg, D.: On the breakdown of solutions to the incompressible Euler equations with free surface boundary. SIAM J. Math. Anal.53(3) 3366–3384 (2021). doi:10.1137/20M1360384

    2021 doi

  23. [30]

    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differential Geometry17(2) 255–306 (1982)

    1982

  24. [32]

    Hu, Z., Luo, C., Yao, Y.: Small scale creation for 2D free boundary Euler equations with surface tension. Ann. PDE 10(2) Paper No. 13, 19 (2024). doi:10.1007/s40818-024-00179-8

    2024 doi

  25. [33]

    Ifrim, M., Pineau, B., Tataru, D., Taylor, M.A.: Sharp Hadamard Local Well-Posedness, Enhanced Uniqueness and Pointwise Continuation Criterion for the Incompressible Free Boundary Euler Equations. Ann. PDE11(1) Paper No. 16 (2025). doi:10.1007/s40818-025-00204-4

    2025 doi

  26. [34]

    Ifrim, M., Tataru, D.: The lifespan of small data solutions in two dimensional capillary water waves. Arch. Ration. Mech. Anal.225(3) 1279–1346 (2017). doi:10.1007/s00205-017-1126-z

    2017 doi

  27. [35]

    Funkcial

    Iguchi, T.: Well-posedness of the initial value problem for capillary-gravity waves. Funkcial. Ekvac.44(2) 219–241 (2001)

    2001

  28. [36]

    Ionescu, A.D., Pusateri, F.: Global solutions for the gravity water waves system in 2d. Invent. Math.199(3) 653–804 (2015). doi:10.1007/s00222-014-0521-4

    2015 doi

  29. [37]

    Jeon, J., Zlatoš, A.: An improved regularity criterion and absence of splash-like singularities for G-SQG patches. Anal. PDE17(3) 1005–1018 (2024). doi:10.2140/apde.2024.17.1005

    2024 doi

  30. [38]

    Julin, V., La Manna, D.A.: A priori estimates for the motion of charged liquid drop: a dynamic approach via free boundary Euler equations. J. Math. Fluid Mech.26(3) Paper No. 48, 83 (2024). doi:10.1007/s00021-024-00883-2

    2024 doi

  31. [39]

    Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41(7) 891–907 (1988). doi:10.1002/cpa.3160410704

    1988 doi

  32. [40]

    Lannes, D.: Well-posedness of the water-waves equations. J. Amer. Math. Soc.18(3) 605–654 (2005). doi:10.1090/ S0894-0347-05-00484-4

    2005

  33. [41]

    Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. of Math. (2)162(1) 109–194 (2005). doi:10.4007/annals.2005.162.109

    2005 doi

  34. [42]

    Luo, C., Zhou, K.: A generalized Beale-Kato-Majda breakdown criterion for the free-boundary problem in Euler equations with surface tension. SIAM J. Math. Anal.56(1) 374–411 (2024). doi:10.1137/23M1563761

    2024 doi

  35. [43]

    Mantegazza, C.: Smooth geometric evolutions of hypersurfaces. Geom. Funct. Anal.12(1) 138–182 (2002). doi: 10.1007/s00039-002-8241-0

    2002 doi

  36. [44]

    Ming, M., Wang, C.: Local well-posedness of the capillary-gravity water waves with acute contact angles. Arch. Ration. Mech. Anal.248(5) Paper No. 72, 71 (2024). doi:10.1007/s00205-024-02019-2

    2024 doi

  37. [45]

    Ogawa, M., Tani, A.: Free boundary problem for an incompressible ideal fluid with surface tension. Math. Models Methods Appl. Sci.12(12) 1725–1740 (2002). doi:10.1142/S0218202502002306

    2002 doi

  38. [46]

    Schweizer, B.: On the three-dimensional Euler equations with a free boundary subject to surface tension. Ann. Inst. H. Poincaré C Anal. Non Linéaire22(6) 753–781 (2005). doi:10.1016/j.anihpc.2004.11.001

    2005 doi

  39. [47]

    Shatah, J., Zeng, C.: Geometry and a priori estimates for free boundary problems of the Euler equation. Comm. Pure Appl. Math.61(5) 698–744 (2008). doi:10.1002/cpa.20213 CLASSIFICATION OF FINITE-TIME BLOW-UP 29

    2008 doi

  40. [48]

    Shatah, J., Zeng, C.: A priori estimates for fluid interface problems. Comm. Pure Appl. Math.61(6) 848–876 (2008). doi:10.1002/cpa.20241

    2008 doi

  41. [49]

    Shatah, J., Zeng, C.: Local well-posedness for fluid interface problems. Arch. Ration. Mech. Anal.199(2) 653–705 (2011). doi:10.1007/s00205-010-0335-5

    2011 doi

  42. [50]

    Springer, Cham (2017)

    Tu, L.W.: Differential geometry, volume 275 of Graduate Texts in Mathematics. Springer, Cham (2017). doi: 10.1007/978-3-319-55084-8. Connections, curvature, and characteristic classes

    2017 doi

  43. [51]

    Wang, C., Zhang, Z., Zhao, W., Zheng, Y.: Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary. Mem. Amer. Math. Soc.270(1318) v + 119 (2021). doi:10.1090/memo/1318

    2021 doi

  44. [52]

    Wang, X.: Global infinite energy solutions for the 2D gravity water waves system. Comm. Pure Appl. Math.71(1) 90–162 (2018). doi:10.1002/cpa.21711

    2018 doi

  45. [53]

    Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc.12(2) 445–495 (1999). doi:10.1090/S0894-0347-99-00290-8

    1999 doi

  46. [54]

    Wu, S.: Global wellposedness of the 3-D full water wave problem. Invent. Math.184(1) 125–220 (2011). doi: 10.1007/s00222-010-0288-1

    2011 doi

  47. [55]

    Zhang, P., Zhang, Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Comm. Pure Appl. Math.61(7) 877–940 (2008). doi:10.1002/cpa.20226 Chengchun Hao State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Ch...

    2008 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.