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Analyticity of Rotational Water Waves

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Elliptic and Parabolic Equations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 119))

Abstract

The aim of this survey is to review some recent results concerning the regularity properties of two-dimensional rotational free-surface flows. It is shown that for large classes of vorticity distributions, the corresponding free water surface together with all streamlines beneath are real-analytic curves. The models considered here include, besides classical periodic water waves of finite depth, solitary waves, waves with infinite depth, capillary waves, and waves over stratified flows. It is also pointed out that the analyticity of the streamlines leads to an intrinsic characterization of symmetric solitary waves with one single crest.

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Notes

  1. 1.

    Helmholtz’s law for two-dimensional flows ensures that the vorticity is constant along C 1-trajectories. This implies particularly that once the vorticity does not vanish in a single point irrotationality is lost forever.

  2. 2.

    The mapping \(\mathcal{H}\) is called a partial hodograph transformation, because its first component is trivial. The full hodograph transformation involves the velocity potential ϕ – the holomorphic conjugate to the stream function ψ, that is \(\varphi + i\psi\) is holomorphic. Note that ϕ is only well defined for irrotational flows. Furthermore, it is worthwhile to mention that the full hodograph transformation is a conformal mapping, in contrast to the partial hodograph transformation \(\mathcal{H}\).

  3. 3.

    By a strong solution of any of the above formulations, we mean a set of functions that satisfies the corresponding system of equations almost everywhere and which fulfils (4), (12), or (15), respectively.

  4. 4.

    In the Sects. 2.2 and 2.3, the Sobolev spaces are defined in the context of λ-periodic functions with respect to the variable x.

  5. 5.

    If \(r=\infty,\) then \(\alpha=1\). In this case, we use the notation \(C^{1+1}(\bar{\Omega}):=W^{2}_\infty(\Omega).\)

  6. 6.

    Given an open subset \(U\subset \mathbb{R}^m\) , \(m\geq 1\), the space \(\mbox{\it BUC}\,^{k} (U)\), \(k\in\mathbb{N},\) consists of all functions which possess bounded uniformly continuous derivatives up to order k. Given \(\alpha\in(0,1)\). \(\mbox{\it BUC}\,^{k+\alpha} (U)\) contains all functions of \(\mbox{\it BUC}\,^{k} (U)\) which have uniformly Hölder continuous derivatives of order k.

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Authors and Affiliations

  1. Institute for Applied Mathematics, Leibniz University of Hannover, 30167, Hannover, Germany

    Joachim Escher & Bogdan-Vasile Matioc

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  1. Joachim Escher
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  2. Bogdan-Vasile Matioc
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Correspondence to Joachim Escher .

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Editors and Affiliations

  1. Institute for Applied Mathematics, Leibniz University of Hannover, Hannover, Germany

    Joachim Escher

  2. Institute for Analysis, Leibniz University of Hannover, Hannover, Germany

    Elmar Schrohe

  3. Department of Mathematics, University of Torino, Torino, Torino, Italy

    Jörg Seiler

  4. Institute for Applied Mathematics, Leibniz University of Hannover, Hannover, Germany

    Christoph Walker

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Escher, J., Matioc, BV. (2015). Analyticity of Rotational Water Waves. In: Escher, J., Schrohe, E., Seiler, J., Walker, C. (eds) Elliptic and Parabolic Equations. Springer Proceedings in Mathematics & Statistics, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-12547-3_5

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