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A Dynamical Systems Approach Towards Isolated Vorticity Regions for Tsunami Background States

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Abstract

We prove the existence of tsunami background states with isolated regions of vorticity beneath a flat free surface and surrounded by still water.

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References

  • Acheson D.J.: Elementary Fluid Dynamics. Oxford University Press, New York (1990)

    MATH  Google Scholar 

  • Alvarez-Samaniego B., Lannes D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485–541 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  • Berger M.S.: Nonlinearity and Functional Analysis. Academic Press, New York (1977)

    MATH  Google Scholar 

  • Bryant, E.: Tsunami: The Underrated Hazard. Springer Praxis Books, Springer Berlin Heidelberg, 2008

  • Constantin A.: On the relevance of soliton theory to tsunami modelling. Wave Motion 46, 420–426 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Constantin A.: On the propagation of tsunami waves, with emphasis on the tsunami of 2004. Discrete Contin. Dyn. Syst. Ser. B 12, 525–537 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 172 (2010)

  • Constantin A., Henry D.: Solitons and tsunamis. Z. Naturforsch. 64a, 65–68 (2009)

    Google Scholar 

  • Constantin A., Johnson R.S.: Modelling tsunamis. J. Phys. A 39, L215–L217 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  • Constantin A., Johnson R.S.: Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis. Fluid Dyn. Res. 40, 175–211 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Constantin, A., Johnson, R.S.: Addendum: propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis. Fluid Dyn. Res. 42, Art. No. 038901 (2010)

  • Constantin A., Strauss W.: Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57, 481–527 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. D. C. Heath and Co., Boston, 1965

  • da Silva A.F.T., Peregrine D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  • Dieudonné J.: Foundations of Modern Analysis. Academic Press, New York (1969)

    MATH  Google Scholar 

  • Drazin P.G., Johnson R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  • Fraenkel L.E., Berger M.S.: A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 13–51 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • Guillemin V., Pollack A.: Differential Topology. Prentice-Hall, New Jersey (1974)

    MATH  Google Scholar 

  • Hammack J.L.: A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60, 769–799 (1973)

    Article  ADS  MATH  Google Scholar 

  • Johnson R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  • Lakshmanan, M.: Integrable nonlinear wave equations and possible connections to tsunami dynamics. In: Tsunami and Nonlinear Waves, pp. 31–49 (ed. A. Kundu). Springer, Berlin, 2007

  • Madsen, P.A., Fuhrman, D.R., Schaeffer, H.A.: On the solitary wave paradigm for tsunamis. J. Geophys. Res. Oceans 113, Art. No. C12012 (2008)

  • Segur, H.: Waves in shallow water with emphasis on the tsunami of 2004. In: Tsunami and Nonlinear Waves, pp. 3–29 (ed. A. Kundu). Springer, Berlin, 2007

  • Segur, H.: Integrable models of waves in shallow water. In: Probability, Geometry and Integrable Systems, pp. 345–371 (eds. M. Pinsky, et al.). Cambridge University Press, Math. Sci. Res. Inst. Publ., 2008

  • Stuhlmeier R.: KdV theory and the Chilean tsunami of 1960. Discrete Contin. Dyn. Syst. Ser. B 12, 623–632 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Zahibo N., Pelinovsky E., Talipova T., Kozelkov A., Kurkin A.: Analytical and numerical study of nonlinear effects at tsunami modeling. Appl. Math. Comput. 174, 795–809 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Vienna, Austria

    Adrian Constantin

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  1. Adrian Constantin
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Correspondence to Adrian Constantin.

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Communicated by A. Bressan

This work was completed with the support of the Vienna Science and Technology Fund WWTF. The author is grateful to the referee for providing constructive and insightful comments that led to an improvement of this paper.

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Constantin, A. A Dynamical Systems Approach Towards Isolated Vorticity Regions for Tsunami Background States. Arch Rational Mech Anal 200, 239–253 (2011). https://doi.org/10.1007/s00205-010-0347-1

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  • DOI: https://doi.org/10.1007/s00205-010-0347-1

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