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Proof of the Stokes conjecture for compressible gravity water waves

Lili Du, Chunlei Yang

TL;DR

This work proves the Stokes conjecture for compressible gravity water waves by developing a Weiss-type monotonicity formula for a quasilinear Bernoulli-type free boundary problem and a nonlinear frequency formula. The authors construct subsonic variational solutions and perform a precise blow-up analysis at stagnation points, identifying the Stokes corner profile as the only nondegenerate limit and classifying degenerate points via a compensated compactness framework. They exclude cusp-type singularities under a strong Bernstein-type growth bound and control horizontal-flat degeneracies through a dedicated frequency analysis, thereby extending the classic incompressible Stokes theory to compressible irrotational flows in two dimensions. The results provide a rigorous variational methodology for singular free-boundary behavior in compressible Euler systems and enhance understanding of extreme gravity waves in fluids with variable density.

Abstract

In 1880, Stokes examined an incompressible irrotational periodic traveling water wave under the influence of gravity and conjectured the existence of an extreme wave with a corner of $120^{\circ}$ at the crest. The first rigorous proof of the conjecture was given by Amick, Fraenkel and Toland, as well as by Plotnikov independently via the Nekrasov integral equation. In the early 2010s, Weiss and Varvarucva revisited the conjecture by applying a new geometric method, which provided an affirmative answer to the conjecture without requiring structural assumptions such as the isolation of the stagnation points, the symmetry and the monotonicity of the free surface that were necessary in the previous works. The main purpose of this paper is to establish the validity of the Stokes conjecture in the context of compressible gravity water waves. More precisely, we prove that a sharp crest forms near each stagnation point of a compressible gravity water wave with an included angle of $120^{\circ}$, which gives a first proof to the compressible counterpart of the classical conjecture by Stokes in 1880. The central aspect of our approach is the discovery of a new monotonicity formula for quasilinear free boundary problems of the Bernoulli-type. Another observation is the introduction of a new nonlinear frequency formula, along with a compensated compactness argument for the compressible Euler system. The developed monotonicity formula enables us to do blow-up analysis at each stagnation point and helps us obtain the singular profile of the free surface near each stagnation points. The degenerate stagnation points can be further analyzed with the help of the compensated compactness argument using the frequency formula.

Proof of the Stokes conjecture for compressible gravity water waves

TL;DR

This work proves the Stokes conjecture for compressible gravity water waves by developing a Weiss-type monotonicity formula for a quasilinear Bernoulli-type free boundary problem and a nonlinear frequency formula. The authors construct subsonic variational solutions and perform a precise blow-up analysis at stagnation points, identifying the Stokes corner profile as the only nondegenerate limit and classifying degenerate points via a compensated compactness framework. They exclude cusp-type singularities under a strong Bernstein-type growth bound and control horizontal-flat degeneracies through a dedicated frequency analysis, thereby extending the classic incompressible Stokes theory to compressible irrotational flows in two dimensions. The results provide a rigorous variational methodology for singular free-boundary behavior in compressible Euler systems and enhance understanding of extreme gravity waves in fluids with variable density.

Abstract

In 1880, Stokes examined an incompressible irrotational periodic traveling water wave under the influence of gravity and conjectured the existence of an extreme wave with a corner of at the crest. The first rigorous proof of the conjecture was given by Amick, Fraenkel and Toland, as well as by Plotnikov independently via the Nekrasov integral equation. In the early 2010s, Weiss and Varvarucva revisited the conjecture by applying a new geometric method, which provided an affirmative answer to the conjecture without requiring structural assumptions such as the isolation of the stagnation points, the symmetry and the monotonicity of the free surface that were necessary in the previous works. The main purpose of this paper is to establish the validity of the Stokes conjecture in the context of compressible gravity water waves. More precisely, we prove that a sharp crest forms near each stagnation point of a compressible gravity water wave with an included angle of , which gives a first proof to the compressible counterpart of the classical conjecture by Stokes in 1880. The central aspect of our approach is the discovery of a new monotonicity formula for quasilinear free boundary problems of the Bernoulli-type. Another observation is the introduction of a new nonlinear frequency formula, along with a compensated compactness argument for the compressible Euler system. The developed monotonicity formula enables us to do blow-up analysis at each stagnation point and helps us obtain the singular profile of the free surface near each stagnation points. The degenerate stagnation points can be further analyzed with the help of the compensated compactness argument using the frequency formula.
Paper Structure (27 sections, 23 theorems, 345 equations, 4 figures)

This paper contains 27 sections, 23 theorems, 345 equations, 4 figures.

Table of Contents

  1. Introduction and setups
  2. Two dimensional compressible gravity water waves
  3. A quasilinear free boundary problem
  4. Subsonic state near the stagnation points
  5. Main results and overview of the approach
  6. State of art
  7. History and background
  8. Notations and preliminaries
  9. Variational solutions and Weiss-type monotonicity formula
  10. Two preparatory identities
  11. A Weiss-type monotonicity formula for quasilinear problem
  12. Stokes corner asymptotics at the stagnation points
  13. Weighted densities
  14. Possible singular profiles in two dimensions
  15. Non-degenerate stagnation points
  16. ...and 12 more sections

Key Result

Lemma 1.1

Let $x^{\circ}=(x_{1}^{\circ},x_{2}^{\circ})\in \partial\{\psi>0\}$ be a subsonic free boundary point, then there exists a small ball $B_{r}(x^{\circ})$ so that $\rho(|\nabla\psi|^{2};x_{2})>\rho_{\mathrm{cr},x_{2}}$ for all $x=(x_{1},x_{2})\in B_{r}(x^{\circ})$.

Figures (4)

  • Figure 1: The limit function $\psi_{0}(R,\theta)$
  • Figure 2: Stokes corner
  • Figure 3: Cusps
  • Figure 4: Horizontal flat singularity

Theorems & Definitions (71)

  • Lemma 1.1
  • Remark 2.1
  • Lemma 3.1
  • Definition 4.1: Subsonic variational solution
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Definition 4.6: Subsonic weak solutions
  • Remark 4.7
  • ...and 61 more