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On Local Singularities in Ideal Potential Flows with Free Surface

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Abstract

Despite important advances in the mathematical analysis of the Euler equations for water waves, especially over the last two decades, it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems. For ideal free-surface flow with zero surface tension and gravity, the authors review existing works that describe “splash singularities”, singular hyperbolic solutions related to jet formation and “flip-through”, and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure. The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation. Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.

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Acknowledgments

The authors are grateful to Sergey Gavriluk for historical references regarding ellipsoidal solutions.

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Authors and Affiliations

  1. Department of Physics and Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Jian-Guo Liu

  2. Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania, PA, 12513, USA

    Robert L. Pego

Authors
  1. Jian-Guo Liu
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  2. Robert L. Pego
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Correspondence to Jian-Guo Liu or Robert L. Pego.

Additional information

Dedicated to Andrew J. Majda on the occasion of his 70th birthday

This work was supported by the National Science Foundation under NSF Research Network Grant RNMS11-07444 (KI-Net), the NSF Grants DMS-1514826, DMS-1812573, DMS-1515400, DMS-1812609 and the Simons Foundation under Grant 395796.

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Liu, JG., Pego, R.L. On Local Singularities in Ideal Potential Flows with Free Surface. Chin. Ann. Math. Ser. B 40, 925–948 (2019). https://doi.org/10.1007/s11401-019-0167-z

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  • DOI: https://doi.org/10.1007/s11401-019-0167-z

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