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Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations

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Pith's one-line read This paper gives the first rigorous construction of traveling bore waves — smooth fronts joining two distinct shear-flow states — for the free-boundary incompressible Navier-Stokes equations in a single shallow layer of fluid, with…

desk verdict First rigorous viscous bores; mostly sound, with a fixable Korn-estimate omission that isn't fatal. read the letter →

arxiv 2505.24562 v1 pith:KRXOB7ZO submitted 2025-05-30 math.AP

Noah Stevenson , Ian Tice This is my paper
classification math.AP MSC 35Q3535C0776D3335B4035J6676A2076L0534C37
keywords borewavestravelingfrontsfreeboundaryNavier-StokesviscoussurfaceshallowwaterheteroclinicorbitsthindomainsStokesproblem
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 single shallow layer of viscous, incompressible fluid with a free surface can support traveling bore waves — fronts that join two different constant heights and shear flows far upstream and downstream — with gravity as the only driving force. This is the first construction of such fronts for the free-boundary Navier-Stokes equations; previously, all known viscous traveling surface waves required an external moving force or stress, and classical results showed inviscid single layers cannot sustain bores. The construction rigorously justifies the shallow-water limit by reducing the traveling PDE system to an ODE for the surface height, exhibiting heteroclinic orbits of that ODE, and then lifting each orbit to an exact Navier-Stokes solution through a fixed-point argument in thin domains. The result covers both 'surging' bores (subcritical, small Froude number) and 'ebbing' bores (supercritical), does not require surface tension, and yields bores of large or small height jump depending on the parameter choice.

What carries the argument

The argument is carried by four coupled mechanisms. First, the parameter-tuning identities (1.4.5) fix the asymptotic heights $0<H_-<H_+<1$ so they are independent of $\epsilon$ and coincide with the zeros of the relative-flux polynomial (1.4.2). Second, the shallow-water reduction replaces the PDE by the Liénard-type ODE $\rho''=F(\rho)-G(\rho)\rho'$ with $\rho=\log H$ and $F$, $G$ as in (1.5.16); when $(g,A)\in C_\iota$ the associated potential $V$ and the sign of $G$ force heteroclinic orbits (solutions connecting the two equilibria $\rho_-$ and $\rho_+$) as proved in Theorem 2.5. Third, a general nonautonomous perturbation theorem (Theorem 2.10) shows these orbits persist under small, time-dependent perturbations, producing the 'bore map' that solves the coupled ODEs as a function of the residual unknowns. Fourth, the thin-domain Stokes problem (3.2.1) with stress boundary conditions and Navier slip is shown to be invertible with norms uniform in $\epsilon$ and Lipschitz in the profile $h$ (Theorem 3.9); with a coupling $r_1,r_3,r_4$ chosen to cancel the adversarial linear terms, the residual PDE becomes a contractive fixed-point problem (Proposition 5.3), using the $\epsilon^{3/2}|\log\epsilon|^9$ and $1/|\log\epsilon|$ smallness of the residual sources.

What would settle it

Compute, numerically or by spectral analysis, the operator norm of the inverse thin-domain Stokes map $(L^\epsilon_h)^{-1}$ in the norms of estimate (3.2.73) for a flat profile $h\equiv1$; if the norm grows like $\epsilon^{-\alpha}$ for any $\alpha>0$, the contraction argument cannot apply. A second check: take $g=8$, the one value excluded from $C_{-1}\cup C_1$ in Lemma 2.2, and numerically integrate the ODE (1.5.15); the appearance of a heteroclinic orbit there would show the parameter sets are not the end of the story, though Theorem 1 itself claims only existence in $C_\iota$, not non-existence outside.

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Extended reading notes

Core claim

Fix viscosity $\mu>0$, slip parameter $a>0$, a sign $\iota\in\{-1,1\}$, a gravity-height pair $(g,A)$ in the explicit sets $C_\iota$ defined in (1.5.2), and surface tension $\sigma\ge0$. The paper proves there is a small-depth threshold $\epsilon_*$ such that for every shallowness parameter $\epsilon\in(0,\epsilon_*)$ the nondimensionalized free-boundary Navier-Stokes system (1.4.8) has a smooth classical bore wave in the sense of Definition 0: a free surface $H+\eta$, a velocity field, and a pressure that solve the system, converge to two distinct shear flows with heights $H_+$ and $H_-$ at opposite infinities, and are surging if $\iota=-1$ and ebbing if $\iota=1$. The leading-order free surface $H$ solves the ODE (1.5.3) with limits $H(\iota x)\to H_\pm$ as $x\to\pm\infty$, and the residual $(\eta,u,p)$ is of size $O(\epsilon)$ in the relevant norms. Corollary 4 converts this into dimensional Eulerian bores for essentially every leading-order wave speed $\gamma$, with the surging/ebbing distinction governed by a Froude number $\mathrm{Fr}^2=\gamma\kappa/(2ga)$.

Load-bearing premise

The whole fixed-point construction rests on the claim that the solution operator for the linearized Stokes problem in the thin layer has norm bounded independently of the layer depth $\epsilon$ (and depends Lipschitz-continuously on the profile); if that norm grew like any positive power of $1/\epsilon$, the residual errors, which are only small like $\epsilon^{3/2}|\log\epsilon|^9$ and $1/|\log\epsilon|$, would not be small enough to close the contraction.

Editorial extensions

If this is right

  • Bores exist for arbitrarily shallow layers, with the free surface, velocity, and pressure differing from their asymptotic shear states by $O(\epsilon)$; the construction yields both 'small' bores ($A$ close to $1$) and 'large' bores with height jump close to $1$ ($A$ close to $0$).
  • As $\epsilon\to0$ the Navier-Stokes bore lies within $O(\epsilon)$ of a traveling-wave solution of the one-dimensional viscous shallow water equations with laminar drag (Corollary 3), a rigorous confirmation of the formal shallow-water limit.
  • In physical units, for any viscosity, horizontal gravity, slip parameter, vertical gravity, and surface tension, every leading-order wave speed $\gamma$ except $\gamma=2ga/\kappa$ admits smooth Eulerian bore waves in sufficiently shallow layers; surging bores occur exactly when the Froude number $\mathrm{Fr}<1$ and ebbing bores when $\mathrm{Fr}>1$ (Corollary 4).
  • Surface tension is not needed: the theorem holds with $\sigma=0$, so a purely gravitational mechanism suffices to sustain nontrivial viscous traveling surface waves.

Reading between the lines

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

  • The single value $g=8$ omitted by the theorem is exactly the critical-Froude line $\mathrm{Fr}=1$; the paper makes no claim there, but a natural conjecture is that critical-speed bores, if they exist, need a different construction or break down at the shallow-water level — a concrete next test.
  • The modular design — heteroclinic germ plus nonautonomous persistence plus thin-domain fixed point — suggests the same machinery could produce other gravity-driven viscous waves the authors flag as open: upstream-traveling bores ($\gamma<0$), purely vertical gravity ($\kappa=0$), or other capillary scaling regimes.
  • The estimate that really carries the proof is the $\epsilon$-uniform invertibility of the thin-domain Stokes operator (Theorem 3.9); an independent proof or numerical verification of that estimate would determine how far the shallow-water construction can be pushed beyond the stated threshold.
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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

2 major / 4 minor

Summary. The paper claims the first rigorous construction of two-dimensional traveling bore wave solutions to the free-boundary incompressible Navier-Stokes equations in a shallow single-layer fluid over an inclined plane. The strategy is to derive, via a formal shallow-water scaling, a one-dimensional Liénard-type ODE for the leading-order free surface height, prove existence of heteroclinic orbits connecting two distinct shear-flow equilibria using phase-plane and energy arguments, then build actual Navier-Stokes bore solutions as small perturbations of these orbits. The perturbation argument combines a nonautonomous heteroclinic persistence theorem, a detailed linear theory for a Stokes operator in ε-thin domains with stress and Navier-slip boundary conditions, a careful ansatz separating ODE and residual unknowns, and a contraction-mapping fixed point. The main theorem asserts existence of smooth surging and ebbing bores for explicit parameter regions C_{-1} ∪ C_1, with surface tension optional and with uniform-in-ε estimates down to a small-depth threshold.

Significance. If the proof is completed, the result would be a substantial advance: it would provide the first construction of nontrivial gravity-driven traveling viscous surface waves with different asymptotic heights, answering a longstanding question highlighted by Rayleigh's inviscid impossibility argument. The paper is self-contained in its main architecture: the heteroclinic orbits are produced from explicit ODE coefficients, the parameter tuning (1.4.5) is explicit, the thin-domain Stokes theory is developed from scratch, and the fixed-point scheme gives quantitative control of the residual. The paper also gives concrete falsifiable predictions in the form of parameter regions and leading-order profiles, and it carefully separates the ODE analysis, the PDE linear theory, and the nonlinear fixed-point argument. The main concern is whether one central estimate in the thin-domain Korn inequality is actually proved with the ε-uniform constants required downstream.

major comments (2)
  1. [§3.1, Proposition 3.6, Eqs. (3.1.54)–(3.1.67)] The proof of the thin-domain Korn estimate is not complete as written. The displayed line (3.1.65) contains the term ε^{-1}||Dψ||^2_{L^2(Ω_{εh})}, which comes directly from the bound (3.1.54) on R_1ψ. Taking square roots in (3.1.65) therefore produces a term ε^{-1/2}||Dψ||_{L^2(Ω_{εh})} in the estimate for ||Rψ12||_{L^2(R)}. This term is absent from the claimed bound (3.1.66), and if it is restored the following line (3.1.67) becomes ||∇ψ|| ≲ ||Dψ|| + ||Dψ||^{1/2}||ψ||^{1/2} + ε||ψ||, where the first term has a coefficient that is not small in ε. The advertised absorption leading to (3.1.37) is therefore not justified by the displayed inequalities. The estimate (3.1.36) may well be true, but the text as written does not prove it, and the ε-uniformity of the Stokes inverse in Theorem 3.9 rests on this step.
  2. [§5.3, Proposition 5.3 and Theorem 5.4] The fixed-point argument makes the ε-uniform invertibility of the Stokes operator load-bearing. The contraction estimates (5.1.22)–(5.1.23) close only because the residuals are of size ε^{3/2}|log ε|^9 and 1/|log ε|; if the inverse operator norms or Lipschitz constants in Theorem 3.9's estimate (3.2.73) contained any positive power of ε^{-1}, these residuals would not be small enough to apply the contraction mapping theorem. Since the proof of that estimate depends directly on the coercivity (3.2.12), which in turn depends on Proposition 3.6, the gap identified in the preceding comment affects the central existence claim of Theorem 1. The authors need either to supply the missing estimate for the ε^{-1/2}||Dψ|| term or to replace the Korn argument with a different proof of the ε-uniform a priori bound.
minor comments (4)
  1. [§2.1, Lemma 2.1 and Lemma 2.2] There are several typos in these lemmas: 'strictly deceasing' should be 'strictly decreasing', and in Lemma 2.2 'of and only if' should be 'if and only if'.
  2. [§3.1, Lemma 3.5, Eq. (3.1.31)] The notation for the average of (1/2)Aψ over the domain Ω_{εh} ∩ Q_ε(x_0) is not fully explicit; the displayed formula should indicate clearly that the integral is normalized by the measure of that set, and the domain of integration should be stated consistently in the bounds that follow.
  3. [§4.3, Definition 4.13] The notation f_j(R,ε;·) is slightly misleading because the maps f_j do not depend on R except through the admissible sets E^R_ε and F^R_ε; renaming the maps or the sets would improve readability.
  4. [§3.2, Eq. (3.2.66)] The long identity for ∂_2p should be checked for missing parentheses or a missing factor of 1/h; as typeset it is difficult to verify by eye, and the derivation is important for the normal-derivative estimate (3.2.67).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the PDE existence is built from independently solved ODE and Stokes problems.

full rationale

The paper's derivation chain is not circular. The main theorem is an existence proof: the ODE profile H is constructed in Section 2 from the sign conditions encoded in C_iota (Lemma 2.2, Proposition 2.4, Theorem 2.5), and the Navier-Stokes solution is then obtained in Section 5 as H plus a residual (eta,u,p) produced by a contraction mapping. The crucial Stokes invertibility (Theorem 3.9) is proved from the thin-domain Korn inequality and a method-of-continuity argument; it is not assumed from the conclusions. The parameter tuning (1.4.5) chooses the O(epsilon^2) speed and flux corrections to make the end-state heights H_+/- epsilon-independent; this is a parameter/domain restriction, not a fit of the target bore profile to the PDE. The sets C_iota are exactly the regions where the shallow-water ODE admits heteroclinic orbits, and the existence of PDE bores is then established independently of the ODE by a fixed point. Corollary 3's shallow-water estimate is a proven bound on the constructed solution, with residual smallness obtained from the contraction estimates (Propositions 5.1-5.3), not assumed via the ansatz. The self-citations to Leoni-Tice [47] and Stevenson-Tice [80,81] are used for auxiliary linear estimates and motivational context; they are not invoked as an unverified premise that forces the central existence result, and no 'uniqueness theorem' from the authors is used to exclude alternatives. The skeptical concern about the Korn estimate (3.1.65)-(3.1.67) is a correctness or proof-gap question, not a circularity question: even if the epsilon-uniformity of Theorem 3.9 were in doubt, the argument would be incomplete rather than circular. No equation is defined in terms of the target solution, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central existence claim rests on well-established PDE and ODE tools (Sobolev theory, elliptic regularity, stable and unstable manifold theorems, Lax-Milgram, method of continuity) plus a set of specific modeling choices: the shallow-water parameter scaling (1.3.1), the identification of the Saint-Venant ODE system as the epsilon to 0 limit via the same authors' [81], and the parameter-tuning identities (1.4.5). The free parameters A, g, and epsilon index the family of bores; none is fitted to data. No new physical entities are introduced.

free parameters (3)
  • A (relative flux / height separation parameter) = 0 < A < 1, freely chosen
    A parameterizes the asymptotic heights H_plus and H_minus via (1.4.4) and the relative flux; the theorem holds for every A in (0,1) with (g,A) in C_iota. It is not fitted to data; it is a free construction parameter.
  • g (nondimensional vertical gravity) = g in the open sets C_1 or C_{-1} defined in (1.5.2)
    The theorem holds for every g in C_1 (including g = 0) or C_{-1}; the parameter is restricted by sign conditions on the damping function G in the Liénard ODE. It is not fitted to data.
  • epsilon (shallowness) = 0 < epsilon < epsilon_star
    Existence is asserted for all sufficiently small epsilon; epsilon_star is a threshold determined by the other parameters, not a fitted constant.
assumptions (8)
  • standard math Standard functional analysis: Sobolev embeddings, Lax-Milgram, Fredholm theory, elliptic regularity for Stokes systems (Agmon-Douglis-Nirenberg), and method of continuity.
    Used throughout Sections 3 through 5; for example Theorem 3.9 combines the method of continuity with ADN regularity, and Proposition 3.7 uses Lax-Milgram and Riesz representation.
  • standard math Standard dynamical systems: stable and unstable manifold theorem, Poincaré-Bendixson, and global flows for compactly supported vector fields.
    Section 2.1, Lemmas 2.3 and 2.4 and Theorem 2.5 rely on Perko [70] and Irwin [39] for invariant manifold theory and Poincaré-Bendixson.
  • standard math Korn-type inequalities in thin domains, adapted from Lewicka-Müller [51] and Lewicka [50] to the unbounded epsilon-thin strip with a small trace term.
    Proposition 3.6 proves the specific estimate (3.1.36) by adapting the gradient replacement method of [51]; the paper notes that the exact result is not in the cited works.
  • domain assumption The free-boundary Navier-Stokes traveling system (1.1.3) with Navier slip, and its flattened reformulation (1.3.7) and final form (1.4.8), is the model; derivation from the time-dependent equations is delegated to [47,41].
    Section 1.1 states the system directly in traveling form and refers to [47,41] for the derivation, so the model itself is taken as the starting point.
  • domain assumption The shallow-water scaling (1.3.1): a = epsilon a, sigma = epsilon sigma, g = g/epsilon, gamma = gamma + epsilon^2 gamma_tilde, with normalization gamma -> 4 and kappa/a -> 4.
    This ansatz defines the epsilon-shallow regime in which the paper works; the main theorem is asserted only for this scaling and no other scaling is analysed.
  • domain assumption The one-dimensional viscous Saint-Venant system (2.1.1) with laminar drag is the epsilon to 0 limit for traveling waves, taken from the same authors' [81, Appendix B].
    Section 2 opens by asserting that (2.1.1) is the limit of (1.4.8) as epsilon -> 0, citing [81, Appendix B]; the rigorous link is established only for the constructed bore family via Corollary 3.
  • ad hoc to paper The parameter-tuning identities (1.4.5) choose gamma and A so that the heights H_plus and H_minus solve the cubic flux identity (1.4.2) for all epsilon > 0.
    Equation (1.4.5) restricts the O(epsilon^2) wave-speed and flux corrections to special values that make the asymptotic heights epsilon-independent; this is a construction choice, not a physical law.
  • ad hoc to paper The solution ansatz (1.5.18) decomposes the full unknown into an ODE-determined leading order part plus an epsilon^2 residual, and the coupling operator is chosen to cancel the adversarial linear operator.
    The success of the contraction argument in Section 5 hinges on this particular decomposition and on the choice of the coupling operator C_epsilon approximately equal to -A_epsilon; it is a proof strategy, not a physical input.

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Pith. "Pith review of Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations." pith.science (2026). https://pith.science/paper/KRXOB7ZO

@misc{pith2026250524562,
  author       = {Pith},
  title        = {Pith review of: Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KRXOB7ZO}},
  note         = {Machine review of arXiv:2505.24562}
}
read the original abstract

We give the first mathematical construction of two-dimensional traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations for a single finite depth layer of constant density fluid. Our construction is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers.

Figures

Figures reproduced from arXiv: 2505.24562 by the authors.

Figure 1
Figure 1. On the left we have the plot of Φ(h) from (1.1.7) when γ > 0, in which case choosing a negative flux constant, Φ∗ ∈ (Φmin, 0), allows for two positive solutions, 0 < h− < h+, that give rise to distinct shear flow solutions ζ± = h±, q± = gh±, and v± = s±e1. On the right we have cartoons of these two shear flow solutions, with the h+ solution above the dotted line and h− below. The arrows plot the velocities with the … view at source ↗
Figure 2
Figure 2. The blue and red shaded regions in (g, A)-space depicted above, labeled C±1 and defined in (1.5.2), correspond to the parameter values for which we are able to produce traveling bore waves in Theorem 1. The waves coming from the blue C1 region are ebbing bores, whereas the waves coming from the red C−1 region are surging bores. The parameter A ∈ (0, 1) on the vertical axis measures the separation of the equilibrium … view at source ↗
Figure 3
Figure 3. Here we present four numerical simulations of the graphs of H, the leading order parts of the free surface components corresponding to our rightward traveling bores constructed by Theorem 1 (with σ = 0). Aspects of the fluid velocity are depicted in the subgraph. The streamlines shown are that of the flow seen by an observer moving with the wave. The color indicates the variation in the curl, which is negative throu… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: We plot the graph of the potential energy V (1.5.17) of the Li´enard system (1.5.15). The states ρ− and ρ+ are the critical points of V . The top dashed line indicates V = 0 and intersects the equipotential states ρ− and ρ⋆. The lower dashed line shows V = V (ρ+) < 0. …
Figure 5
Figure 5. Figure 5: Shown here are a streamline plots of the vector field Φ from (2.1.2) superimposed on the trapping region R (from the fourth item of Lemma 2.1) in green, with the boundary of R shown in red. Both plots obey (µ, a) = (2, 1). The plot on the left corresponds to parameter …

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Reference graph

Works this paper leans on

93 extracted references · 77 canonical work pages · cited by 1 Pith paper

  1. [1]

    Y. A. Abramovich and C. D. Aliprantis.An Invitation to Operator Theory, volume 50 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002

    2002

  2. [2]

    Agmon, A

    S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II.Comm. Pure Appl. Math., 17:35–92, 1964

    1964

  3. [3]

    C. J. Amick, L. E. Fraenkel, and J. F. Toland. On the Stokes conjecture for the wave of extreme form.Acta Math., 148:193–214, 1982

    1982

  4. [4]

    C. J. Amick and J. F. Toland. On periodic water-waves and their convergence to solitary waves in the long-wave limit.Philos. Trans. Roy. Soc. London Ser. A, 303(1481):633–669, 1981

    1981

  5. [5]

    C. J. Amick and J. F. Toland. On solitary water-waves of finite amplitude.Arch. Rational Mech. Anal., 76(1):9–95, 1981

    1981

  6. [6]

    C. J. Amick and R. E. L. Turner. Small internal waves in two-fluid systems.Arch. Rational Mech. Anal., 108(2):111–139, 1989

    1989

  7. [7]

    Barreira and C

    L. Barreira and C. Valls.Stability of Nonautonomous Differential Equations, volume 1926 ofLecture Notes in Mathematics. Springer, Berlin, 2008

    1926

  8. [8]

    J. T. Beale. The existence of solitary water waves.Comm. Pure Appl. Math., 30(4):373–389, 1977

    1977

Show all 93 references
  1. [9]

    Bouchut, A

    F. Bouchut, A. Mangeney-Castelnau, B. Perthame, and J.-P. Vilotte. A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows.C. R. Math. Acad. Sci. Paris, 336(6):531–536, 2003

    2003

  2. [10]

    Bouchut and M

    F. Bouchut and M. Westdickenberg. Gravity driven shallow water models for arbitrary topography.Commun. Math. Sci., 2(3):359–389, 2004

    2004

  3. [11]

    Boutounet, L

    M. Boutounet, L. Chupin, P. Noble, and J. P. Vila. Shallow water viscous flows for arbitrary topography.Commun. Math. Sci., 6(1):29–55, 2008

    2008

  4. [12]

    Boyer and P

    F. Boyer and P. Fabrie.Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, volume 183 ofApplied Mathematical Sciences. Springer, New York, 2013

    2013

  5. [13]

    D. Bresch. Shallow-water equations and related topics. InHandbook of Differential Equations: Evolutionary Equations. Vol. V, Handb. Differ. Equ., pages 1–104. Elsevier/North-Holland, Amsterdam, 2009

    2009

  6. [14]

    Bresch and P

    D. Bresch and P. Noble. Mathematical derivation of viscous shallow-water equations with zero surface tension.Indiana Univ. Math. J., 60(4):1137–1169, 2011

    2011

  7. [15]

    Brownfield and H

    J. Brownfield and H. Q. Nguyen. Slowly traveling gravity waves for Darcy flow: existence and stability of large waves.Comm. Math. Phys., 405(10):Paper No. 222, 25, 2024

    2024

  8. [16]

    B¨ uhler, J

    O. B¨ uhler, J. Shatah, and S. Walsh. Steady water waves in the presence of wind.SIAM J. Math. Anal., 45(4):2182–2227, 2013

    2013

  9. [17]

    B¨ uhler, J

    O. B¨ uhler, J. Shatah, S. Walsh, and C. Zeng. On the wind generation of water waves.Arch. Ration. Mech. Anal., 222(2):827– 878, 2016

    2016

  10. [18]

    R. M. Chen, S. Walsh, and M. H. Wheeler. Existence and qualitative theory for stratified solitary water waves.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35(2):517–576, 2018

    2018

  11. [19]

    R. M. Chen, S. Walsh, and M. H. Wheeler. Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity.Arch. Ration. Mech. Anal., 234(2):595–633, 2019

    2019

  12. [20]

    R. M. Chen, S. Walsh, and M. H. Wheeler. Large-amplitude internal fronts in two-fluid systems.C. R. Math. Acad. Sci. Paris, 358(9-10):1073–1083, 2020

    2020

  13. [21]

    R. M. Chen, S. Walsh, and M. H. Wheeler. Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics.Nonlinearity, 35(4):1927–1985, 2022

    1927

  14. [22]

    R. M. Chen, S. Walsh, and M. H. Wheeler. Global bifurcation for monotone fronts of elliptic equations.Journal of the European Mathematical Society, 2024

    2024

  15. [23]

    Y. Cho, J. D. Diorio, T. R. Akylas, and J. H. Duncan. Resonantly forced gravity–capillary lumps on deep water. Part 2. Theoretical model.J. Math. Fluid Mech., 672:288–306, 2011

    2011

  16. [24]

    Constantin and W

    A. Constantin and W. Strauss. Exact steady periodic water waves with vorticity.Comm. Pure Appl. Math., 57(4):481–527, 2004

    2004

  17. [25]

    Cornish.Ocean Waves and Kindred Geophysical Phenomena

    V. Cornish.Ocean Waves and Kindred Geophysical Phenomena. Cambridge University Press, 1934

    1934

  18. [26]

    Dias and G

    F. Dias and G. Iooss. Water-waves as a spatial dynamical system. InHandbook of Mathematical Fluid Dynamics, Vol. II, pages 443–499. North-Holland, Amsterdam, 2003

    2003

  19. [27]

    J. D. Diorio, Y. Cho, J. H. Duncan, and T. R. Akylas. Resonantly forced gravity–capillary lumps on deep water. Part 1. Experiments.J. Math. Fluid Mech., 672:268–287, 2011

    2011

  20. [28]

    Ehrnstr¨ om, S

    M. Ehrnstr¨ om, S. Walsh, and C. Zeng. Smooth stationary water waves with exponentially localized vorticity.J. Eur. Math. Soc. (JEMS), 25(3):1045–1090, 2023

    2023

  21. [29]

    H. Favre. ´Etude Th´ eorique et Exp´ erimentale des Ondes de Translation dans les Canaux D´ ecouverts. Dunod, Paris, 1935

    1935

  22. [30]

    Fern´ andez Bonder, S

    J. Fern´ andez Bonder, S. Mart ´ ınez, and J. D. Rossi. The behavior of the best Sobolev trace constant and extremals in thin domains.J. Differential Equations, 198(1):129–148, 2004

    2004

  23. [31]

    K. O. Friedrichs and D. H. Hyers. The existence of solitary waves.Comm. Pure Appl. Math., 7:517–550, 1954

    1954

  24. [32]

    Gerbeau and B

    J.-F. Gerbeau and B. Perthame. Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B, 1(1):89–102, 2001. TRA VELING BORE W A VES 65

    2001

  25. [33]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger.Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer- Verlag, Berlin, 2001. Reprint of the 1998 edition

    2001

  26. [34]

    M. D. Groves. Steady water waves.J. Nonlinear Math. Phys., 11(4):435–460, 2004

    2004

  27. [35]

    M. D. Groves and E. Wahl´ en. Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity.Phys. D, 237(10-12):1530–1538, 2008

    2008

  28. [36]

    S. V. Haziot, V. M. Hur, W. A. Strauss, J. F. Toland, E. Wahl´ en, S. Walsh, and M. H. Wheeler. Traveling water waves—the ebb and flow of two centuries.Quart. Appl. Math., 80(2):317–401, 2022

    2022

  29. [37]

    S. V. Haziot and M. H. Wheeler. Large-amplitude steady solitary water waves with constant vorticity.Arch. Ration. Mech. Anal., 247(2):Paper No. 27, 49, 2023

    2023

  30. [38]

    V. M. Hur. Exact solitary water waves with vorticity.Arch. Ration. Mech. Anal., 188(2):213–244, 2008

    2008

  31. [39]

    M. C. Irwin.Smooth Dynamical Systems, volume 17 ofAdvanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, 2001

    2001

  32. [40]

    Keady and J

    G. Keady and J. Norbury. On the existence theory for irrotational water waves.Math. Proc. Cambridge Philos. Soc., 83(1):137– 157, 1978

    1978

  33. [41]

    Koganemaru and I

    J. Koganemaru and I. Tice. Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations.J. Funct. Anal., 285(7):Paper No. 110057, 75, 2023

    2023

  34. [42]

    Koganemaru and I

    J. Koganemaru and I. Tice. Traveling wave solutions to the free boundary incompressible Navier-Stokes equations with Navier boundary conditions.J. Differential Equations, 411:381–437, 2024

    2024

  35. [43]

    J. P. Krasovski ˘ ı. On the theory of steady-state waves of finite amplitude.ˇZ. Vyˇ cisl. Mat i Mat. Fiz., 1:836–855, 1961

    1961

  36. [44]

    D. Lannes. Modeling shallow water waves.Nonlinearity, 33(5):R1–R57, 2020

    2020

  37. [45]

    Lavrentiev

    M. Lavrentiev. On the theory of long waves (1943).A contribution to the theory of long waves, pages 3–50, 1947

    1943

  38. [46]

    P. D. Lax.Functional Analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002

    2002

  39. [47]

    Leoni and I

    G. Leoni and I. Tice. Traveling wave solutions to the free boundary incompressible Navier-Stokes equations.Comm. Pure Appl. Math., 76(10):2474–2576, 2023

    2023

  40. [48]

    C. D. Levermore and M. Sammartino. A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity, 14(6):1493–1515, 2001

    2001

  41. [49]

    Levi-Civita

    T. Levi-Civita. Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda.Rendiconti della Accademia dei Lincei, 33:141–150, 1924

    1924

  42. [50]

    Lewicka.Calculus of Variations on Thin Prestressed Films—Asymptotic Methods in Elasticity, volume 101 ofProgress in Nonlinear Differential Equations and their Applications

    M. Lewicka.Calculus of Variations on Thin Prestressed Films—Asymptotic Methods in Elasticity, volume 101 ofProgress in Nonlinear Differential Equations and their Applications. Birkh¨ auser/Springer, Cham, 2023

    2023

  43. [51]

    Lewicka and S

    M. Lewicka and S. M¨ uller. The uniform Korn-Poincar´ e inequality in thin domains.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 28(3):443–469, 2011

    2011

  44. [52]

    Liska, L

    R. Liska, L. Margolin, and B. Wendroff. Nonhydrostatic two-layer models of incompressible flow.Comput. Math. Appl., 29(9):25–37, 1995

    1995

  45. [53]

    N. I. Makarenko. Smooth bore in a two-layer fluid. InFree Boundary Problems in Continuum Mechanics (Novosibirsk, 1991), volume 106 ofInternat. Ser. Numer. Math., pages 195–204. Birkh¨ auser, Basel, 1992

    1991

  46. [54]

    F. Marche. Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects.Eur. J. Mech. B Fluids, 26(1):49–63, 2007

    2007

  47. [55]

    Maruˇ si´ c

    S. Maruˇ si´ c. Asymptotic behaviour of sharp Sobolev constants in thin domains.J. Math. Anal. Appl., 272(2):575–581, 2002

    2002

  48. [56]

    C. Mascia. A dive into shallow water.Riv. Math. Univ. Parma (N.S.), 1(1):77–149, 2010

    2010

  49. [57]

    Masnadi and J

    N. Masnadi and J. H. Duncan. The generation of gravity–capillary solitary waves by a pressure source moving at a trans-critical speed.J. Fluid Mech., 810:448–474, 2017

    2017

  50. [58]

    J. B. McLeod. The Stokes and Krasovskii conjectures for the wave of greatest height.Stud. Appl. Math., 98(4):311–333, 1997

    1997

  51. [59]

    A. Mielke. Reduction of quasilinear elliptic equations in cylindrical domains with applications.Math. Methods Appl. Sci., 10(1):51–66, 1988

    1988

  52. [60]

    A. Mielke. Homoclinic and heteroclinic solutions in two-phase flow. InProceedings of the IUTAM/ISIMM Symposium on Structure and Dynamics of Nonlinear Waves in Fluids (Hannover, 1994), volume 7 ofAdv. Ser. Nonlinear Dynam., pages 353–362. World Sci. Publ., River Edge, NJ, 1995

    1994

  53. [61]

    C. L. M. H. Navier. M´ emoire sur les lois du mouvement des fluides.M´ em. Acad. R. Sci. Inst. France, 6(1823):389–440, 1823

  54. [62]

    A. I. Nekrasov. On steady waves.Izvestiya Ivanovo-Voznesensky Polytechnic Institute, 3, 1921

    1921

  55. [63]

    C. Neto, D. R. Evans, E. Bonaccurso, H.-J. Butt, and V. S. J. Craig. Boundary slip in Newtonian liquids: a review of experimental studies.Rep. Progr. Phys., 68(12):2859, 2005

    2005

  56. [64]

    H. Q. Nguyen. Large traveling capillary-gravity waves for Darcy flow. 2023. Preprint, arXiv:2311.01299

    2023

  57. [65]

    H. Q. Nguyen and I. Tice. Traveling wave solutions to the one-phase Muskat problem: existence and stability.Arch. Ration. Mech. Anal., 248(1):Paper No. 5, 58, 2024

    2024

  58. [66]

    A. Oron, S. H. Davis, and S. G. Bankoff. Long-scale evolution of thin liquid films.Rev. Mod. Phys., 69:931–980, Jul 1997

    1997

  59. [67]

    Park and Y

    B. Park and Y. Cho. Experimental observation of gravity–capillary solitary waves generated by a moving air suction.J. Math. Fluid Mech., 808:168–188, 2016

    2016

  60. [68]

    Park and Y

    B. Park and Y. Cho. Two-dimensional gravity–capillary solitary waves on deep water: generation and transverse instability. J. Math. Fluid Mech., 834:92–124, 2018

    2018

  61. [69]

    R. L. Pego and S.-M. Sun. Asymptotic linear stability of solitary water waves.Arch. Ration. Mech. Anal., 222(3):1161–1216, 2016. 66 NOAH STEVENSON AND IAN TICE

    2016

  62. [70]

    Perko.Differential Equations and Dynamical Systems, volume 7 ofTexts in Applied Mathematics

    L. Perko.Differential Equations and Dynamical Systems, volume 7 ofTexts in Applied Mathematics. Springer-Verlag, New York, third edition, 2001

    2001

  63. [71]

    P. I. Plotnikov. Proof of the Stokes conjecture in the theory of surface waves.Stud. Appl. Math., 108(2):217–244, 2002. Translated from Dinamika Sploshn. Sredy No. 57 (1982), 41–76 [MR0752600 (85f:76036)]

    1982

  64. [72]

    P. I. Plotnikov and J. F. Toland. Convexity of Stokes waves of extreme form.Arch. Ration. Mech. Anal., 171(3):349–416, 2004

    2004

  65. [73]

    Rayleigh

    L. Rayleigh. On the theory of long waves and bores.Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 90(619):324–328, 1914

    1914

  66. [74]

    Reed and B

    M. Reed and B. Simon.Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York-London, 1972

    1972

  67. [75]

    J. S. Russell. Report on waves. In14th Meeting of the British Association for the Advancement of Science, pages 311–390, 1844

  68. [76]

    Stevenson

    N. Stevenson. Periodic gravity-capillary roll wave solutions to the inclined viscous shallow water equations in two dimensions. SIAM J. Math. Anal., 57(2):1342–1369, 2025

    2025

  69. [77]

    Stevenson and I

    N. Stevenson and I. Tice. Traveling wave solutions to the multilayer free boundary incompressible Navier-Stokes equations. SIAM J. Math. Anal., 53(6):6370–6423, 2021

    2021

  70. [78]

    Stevenson and I

    N. Stevenson and I. Tice. The traveling wave problem for the shallow water equations: well-posedness and the limits of vanishing viscosity and surface tension. 2023. Preprint, arXiv:2311.00160

    2023 arXiv

  71. [79]

    Stevenson and I

    N. Stevenson and I. Tice. Well-posedness of the traveling wave problem for the free boundary compressible Navier-Stokes equations. 2023. Preprint, arXiv:2301.00773

    2023 arXiv

  72. [80]

    Stevenson and I

    N. Stevenson and I. Tice. Well-posedness of the stationary and slowly traveling wave problems for the free boundary incom- pressible Navier-Stokes equations.J. Funct. Anal., 287(11):Paper No. 110617, 85, 2024

    2024

  73. [81]

    Stevenson and I

    N. Stevenson and I. Tice. Stationary wave solutions to two dimensional viscous shallow water equations: theory of small and large solutions. 2025. Preprint, arXiv:2502.11899

    2025 arXiv

  74. [82]

    G. G. Stokes. On the theory of oscillatory waves.Trans. Cambridge Philos. Soc., 8:441–455, 1847

  75. [83]

    W. A. Strauss. Steady water waves.Bull. Amer. Math. Soc. (N.S.), 47(4):671–694, 2010

    2010

  76. [84]

    Temam and M

    R. Temam and M. Ziane. Navier-Stokes equations in three-dimensional thin domains with various boundary conditions.Adv. Differential Equations, 1(4):499–546, 1996

    1996

  77. [85]

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

    1978

  78. [86]

    J. F. Toland. Stokes waves.Topol. Methods Nonlinear Anal., 7(1):1–48, 1996

    1996

  79. [87]

    Wahl´ en

    E. Wahl´ en. Steady periodic capillary-gravity waves with vorticity.SIAM J. Math. Anal., 38(3):921–943, 2006

    2006

  80. [88]

    Wahl´ en

    E. Wahl´ en. Steady periodic capillary waves with vorticity.Ark. Mat., 44(2):367–387, 2006

    2006

  81. [89]

    S. Walsh. Stratified steady periodic water waves.SIAM J. Math. Anal., 41(3):1054–1105, 2009

    2009

  82. [90]

    S. Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation.Discrete Contin. Dyn. Syst., 34(8):3241–3285, 2014

    2014

  83. [91]

    S. Walsh. Steady stratified periodic gravity waves with surface tension II: global bifurcation.Discrete Contin. Dyn. Syst., 34(8):3287–3315, 2014

    2014

  84. [92]

    M. H. Wheeler. Large-amplitude solitary water waves with vorticity.SIAM J. Math. Anal., 45(5):2937–2994, 2013

    2013

  85. [93]

    M. H. Wheeler. Solitary water waves of large amplitude generated by surface pressure.Arch. Ration. Mech. Anal., 218(2):1131– 1187, 2015. Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Email address, N. Stevenson:stevenson@princeton.edu Department of ...

    2015

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Reviewed August 7, 2026 · model on record in the stance chip above.