On the existence of extreme waves and the Stokes conjecture with vorticity

@article{Varvaruca2007OnTE,
  title={On the existence of extreme waves and the Stokes conjecture with vorticity},
  author={Eugen Varvaruca},
  journal={Journal of Differential Equations},
  year={2007},
  volume={246},
  pages={4043-4076}
}
  • E. Varvaruca
  • Published 15 July 2007
  • Mathematics
  • Journal of Differential Equations
View PDF on arXiv
66 Citations
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66 Citations

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References

SHOWING 1-10 OF 44 REFERENCES

On Some Properties of Traveling Water Waves with Vorticity

  • E. Varvaruca
  • Mathematics, Environmental Science
    SIAM J. Math. Anal.
  • 2008
We prove that for a large class of vorticity functions the crest of a corresponding travelling water wave is necessarily a point of maximal horizontal velocity. We also show that for waves with…

Global Bifurcation Theory of Deep-Water Waves with Vorticity

The classical deep-water wave problem is to find a periodic traveling wave with a free surface of infinite depth. The main result is the construction of a global connected set of rotational solutions…

Steep solitary waves in water of finite depth with constant vorticity

Solitary waves with constant vorticity in water of finite depth are calculated numerically by a boundary integral equation method. Previous calculations are confirmed and extended. It is shown that…

Periodic waves with constant vorticity in water of infinite depth

Periodic waves propagating at a constant velocity at the surface of a fluid with constant vorticity in water of infinite depth are considered. The problem is solved numerically by a…

Exact steady periodic water waves with vorticity

We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two‐dimensional inviscid periodic…

On the existence theory for irrotational water waves

  • G. KeadyJ. Norbury
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1978
Abstract This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the…

On the existence of a wave of greatest height and Stokes’s conjecture

  • J. Toland
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1978
It is shown that there exists a solution of Nekrasov’s integral equation which corresponds to the existence of a wave of greatest height and of permanent form moving on the surface of an…

Steep, steady surface waves on water of finite depth with constant vorticity

Two-dimensional steady surface waves on a shearing flow are computed for the special case where the flow has uniform vorticity, i.e. in the absence of waves the velocity varies linearly with height.…

Steady Deep‐Water Waves on a Linear Shear Current

The behavior of steady, periodic, deep-water gravity waves on a linear shear current is investigated. A weakly nonlinear approximation for the small amplitude waves is constructed via a variational…

A Constructive Existence Proof for the Extreme Stokes Wave

Stokes conjectured in 1880 that an extreme gravity wave on water (or ‘wave of greatest height’) exists, has sharp crests of included angle 2π/3 and has a boundary that is convex between successive…