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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 and 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), 193-214.

[2]

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

[3]

Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.

[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), 807-821.

[5]

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

[6]

Michael Selwyn Longuet-Higgins, On the forming of sharp corners at a free surface, Proc. R. Soc. Lond. A, 371 (1980), 453-478.

[7]

Michael Selwyn Longuet-Higgins, On the overturning of gravity waves, Proc. R. Soc. Lond. A, 376 (1981), 377-400.

[8]

P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves," Translations of Mathematical Monographs, 133, American Mathematical Society, Providence, RI, 1994.

[9]

D. Howell Peregrine, Breaking waves on beaches, Ann. Rev. Fluid Mech., 15 (1983), 149-178.

[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), 2541-2558.

[11]

R. Thom, "Structural Stability and Morphogenesis," Benjamin, 1975.

[12]

Milton Van Dyke, "An Album of Fluid Motion," Parabolic Press, 10th, 1982.

[13]

Gerald Bereford Whitham, Variational methods and applications to water waves, Hyperbolic equations and waves (Rencontre, Battelle Res. Inst., Seallte, WA, 1968), Springer, Berlin, 1970, 153-172.

[14]

Gerald Beresford Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original. Pure and Applied Mathematics, Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.

[15]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130 (1997), 39-72.

[16]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.

[17]

Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.

[18]

E. C. Zeeman, Breaking of waves, Warwick Symposium on Differential Equations and Dynamical Systems, Lecture notes in Mathematics, 206 (1971), 2-6.

[19]

Antoni Zygmund, "Trigonometric Series," Volume 2, Cambridge Mathematics Library, Cambridge University Press, 1968.

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), 193-214.

[2]

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

[3]

Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.

[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), 807-821.

[5]

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

[6]

Michael Selwyn Longuet-Higgins, On the forming of sharp corners at a free surface, Proc. R. Soc. Lond. A, 371 (1980), 453-478.

[7]

Michael Selwyn Longuet-Higgins, On the overturning of gravity waves, Proc. R. Soc. Lond. A, 376 (1981), 377-400.

[8]

P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves," Translations of Mathematical Monographs, 133, American Mathematical Society, Providence, RI, 1994.

[9]

D. Howell Peregrine, Breaking waves on beaches, Ann. Rev. Fluid Mech., 15 (1983), 149-178.

[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), 2541-2558.

[11]

R. Thom, "Structural Stability and Morphogenesis," Benjamin, 1975.

[12]

Milton Van Dyke, "An Album of Fluid Motion," Parabolic Press, 10th, 1982.

[13]

Gerald Bereford Whitham, Variational methods and applications to water waves, Hyperbolic equations and waves (Rencontre, Battelle Res. Inst., Seallte, WA, 1968), Springer, Berlin, 1970, 153-172.

[14]

Gerald Beresford Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original. Pure and Applied Mathematics, Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.

[15]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130 (1997), 39-72.

[16]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.

[17]

Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.

[18]

E. C. Zeeman, Breaking of waves, Warwick Symposium on Differential Equations and Dynamical Systems, Lecture notes in Mathematics, 206 (1971), 2-6.

[19]

Antoni Zygmund, "Trigonometric Series," Volume 2, Cambridge Mathematics Library, Cambridge University Press, 1968.

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