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July  2012, 11(4): 1465-1474. doi: 10.3934/cpaa.2012.11.1465

On the formation of singularities for surface water waves

Vera Mikyoung Hur 1,

1. 

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States

Received  June 2011 Revised  July 2011 Published  January 2012

A Burgers equation with fractional dispersion is proposed to model waves on the moving surface of a two-dimensional, infinitely deep water under the influence of gravity. For a certain class of initial data, the solution is shown to blow up in finite time.
Keywords: Surface water waves, nonlinear nonlocal equations, blowup., fractional dispersion.
Mathematics Subject Classification: 35A20, 76B15, 35Q3.
Citation: Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465
References:
[1]

C. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form,, Acta Math., 148 (1982), 193.   Google Scholar

[2]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Splash singularity for water waves,, preprint, (2011).   Google Scholar

[3]

Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.   Google Scholar

[4]

H. Dong, D. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations,, Indiana Univ. Math. J., 58 (2009), 807.   Google Scholar

[5]

Vera Mikyoung Hur, Gain of regularity for water waves with surface tension: a model equation,, preprint, (2011).   Google Scholar

[6]

Michael Selwyn Longuet-Higgins, On the forming of sharp corners at a free surface,, Proc. R. Soc. Lond. A, 371 (1980), 453.   Google Scholar

[7]

Michael Selwyn Longuet-Higgins, On the overturning of gravity waves,, Proc. R. Soc. Lond. A, 376 (1981), 377.   Google Scholar

[8]

P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves,", Translations of Mathematical Monographs, (1994).   Google Scholar

[9]

D. Howell Peregrine, Breaking waves on beaches,, Ann. Rev. Fluid Mech., 15 (1983), 149.   Google Scholar

[10]

J.-B. Song and M. L. Banner, On determining the onset and strength of breaking for deep water waves. Part I: Unforced irrotational wave groups,, Journal of Physical Oceanography, 32 (2002), 2541.   Google Scholar

[11]

R. Thom, "Structural Stability and Morphogenesis,", Benjamin, (1975).   Google Scholar

[12]

Milton Van Dyke, "An Album of Fluid Motion,", Parabolic Press, (1982).   Google Scholar

[13]

Gerald Bereford Whitham, Variational methods and applications to water waves,, Hyperbolic equations and waves (Rencontre, (1968), 153.   Google Scholar

[14]

Gerald Beresford Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original. Pure and Applied Mathematics, (1974).   Google Scholar

[15]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D,, Invent. Math., 130 (1997), 39.   Google Scholar

[16]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D,, J. Amer. Math. Soc., 12 (1999), 445.   Google Scholar

[17]

Sijue Wu, Almost global wellposedness of the 2-D full water wave problem,, Invent. Math., 177 (2009), 45.   Google Scholar

[18]

E. C. Zeeman, Breaking of waves,, Warwick Symposium on Differential Equations and Dynamical Systems, 206 (1971), 2.   Google Scholar

[19]

Antoni Zygmund, "Trigonometric Series,", Volume 2, (1968).   Google Scholar

show all references

References:
[1]

C. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form,, Acta Math., 148 (1982), 193.   Google Scholar

[2]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Splash singularity for water waves,, preprint, (2011).   Google Scholar

[3]

Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.   Google Scholar

[4]

H. Dong, D. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations,, Indiana Univ. Math. J., 58 (2009), 807.   Google Scholar

[5]

Vera Mikyoung Hur, Gain of regularity for water waves with surface tension: a model equation,, preprint, (2011).   Google Scholar

[6]

Michael Selwyn Longuet-Higgins, On the forming of sharp corners at a free surface,, Proc. R. Soc. Lond. A, 371 (1980), 453.   Google Scholar

[7]

Michael Selwyn Longuet-Higgins, On the overturning of gravity waves,, Proc. R. Soc. Lond. A, 376 (1981), 377.   Google Scholar

[8]

P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves,", Translations of Mathematical Monographs, (1994).   Google Scholar

[9]

D. Howell Peregrine, Breaking waves on beaches,, Ann. Rev. Fluid Mech., 15 (1983), 149.   Google Scholar

[10]

J.-B. Song and M. L. Banner, On determining the onset and strength of breaking for deep water waves. Part I: Unforced irrotational wave groups,, Journal of Physical Oceanography, 32 (2002), 2541.   Google Scholar

[11]

R. Thom, "Structural Stability and Morphogenesis,", Benjamin, (1975).   Google Scholar

[12]

Milton Van Dyke, "An Album of Fluid Motion,", Parabolic Press, (1982).   Google Scholar

[13]

Gerald Bereford Whitham, Variational methods and applications to water waves,, Hyperbolic equations and waves (Rencontre, (1968), 153.   Google Scholar

[14]

Gerald Beresford Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original. Pure and Applied Mathematics, (1974).   Google Scholar

[15]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D,, Invent. Math., 130 (1997), 39.   Google Scholar

[16]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D,, J. Amer. Math. Soc., 12 (1999), 445.   Google Scholar

[17]

Sijue Wu, Almost global wellposedness of the 2-D full water wave problem,, Invent. Math., 177 (2009), 45.   Google Scholar

[18]

E. C. Zeeman, Breaking of waves,, Warwick Symposium on Differential Equations and Dynamical Systems, 206 (1971), 2.   Google Scholar

[19]

Antoni Zygmund, "Trigonometric Series,", Volume 2, (1968).   Google Scholar

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