July  2012, 11(4): 1431-1438. doi: 10.3934/cpaa.2012.11.1431

A note on uniqueness and compact support of solutions in a recent model for tsunami background flows

1. 

Nordbergstrasse 15, UZA 2, 2A 287, 1090 Wien, Austria

Received  May 2011 Revised  September 2011 Published  January 2012

We present an elementary proof of uniqueness for solutions of an initial value problem which is not Lipschitz continuous, generalizing a technique employed in [20]. This approach can be applied for a wide class of vorticity functions in the context of [6], where, departing from a recent model for the evolution of tsunami waves developed in [10], the possibility of modelling background ows with isolated regions of vorticity is rigorously established.
Citation: Anna Geyer. A note on uniqueness and compact support of solutions in a recent model for tsunami background flows. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1431-1438. doi: 10.3934/cpaa.2012.11.1431
References:
[1]

G. K. Batchelor, "An Introduction to Fluid Dynamics,", Cambridge University Press, (1967). Google Scholar

[2]

E. Bryant, "Tsunami: The Underrated Hazard,", Springer Praxis Books, (2008). Google Scholar

[3]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. Google Scholar

[4]

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

[5]

A. Constantin, On the relevance of soliton theory to tsunami modelling,, Wave Motion, 46 (2009), 420. Google Scholar

[6]

A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states,, Arch. Ration. Mech. Anal., 200 (2011), 239. Google Scholar

[7]

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

[8]

A. Constantin and D. Henry, Solitons and tsunamis,, Z. Naturforsch., 64a (2009), 65. Google Scholar

[9]

A. Constantin and R. S. Johnson, Modelling tsunamis,, J. Phys. A, 39 (2006). Google Scholar

[10]

A. Constantin and R. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,, Fluid Dynamics Research, 40 (2008). Google Scholar

[11]

A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves,, J. Nonl. Math. Phys., 15 (2008), 58. Google Scholar

[12]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. Google Scholar

[13]

W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,", D. C. Heath and Co., (1965). Google Scholar

[14]

W. Craig, Surface water waves and tsunamis,, J. Dynam. Differential Equations, 18 (2006), 525. Google Scholar

[15]

A. Geyer, On some background flows for tsunami waves,, J. Math. Fluid Mech.., (). doi: DOI 10.1007/s00021-011-0055-0. Google Scholar

[16]

J. Hale, "Ordinary Differential Equations,", Wiley, (1969). Google Scholar

[17]

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

[18]

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

[19]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31. Google Scholar

[20]

O. Mustafa, On the uniqueness of flow in a recent tsunami model,, Applicable Analysis, (2011). doi: doi: 10.1080/00036811.2011.569499. Google Scholar

[21]

H. Segur, Waves in shallow water with emphasis on the tsunami of 2004,, in, (2007), 3. Google Scholar

[22]

R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 623. Google Scholar

show all references

References:
[1]

G. K. Batchelor, "An Introduction to Fluid Dynamics,", Cambridge University Press, (1967). Google Scholar

[2]

E. Bryant, "Tsunami: The Underrated Hazard,", Springer Praxis Books, (2008). Google Scholar

[3]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. Google Scholar

[4]

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

[5]

A. Constantin, On the relevance of soliton theory to tsunami modelling,, Wave Motion, 46 (2009), 420. Google Scholar

[6]

A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states,, Arch. Ration. Mech. Anal., 200 (2011), 239. Google Scholar

[7]

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

[8]

A. Constantin and D. Henry, Solitons and tsunamis,, Z. Naturforsch., 64a (2009), 65. Google Scholar

[9]

A. Constantin and R. S. Johnson, Modelling tsunamis,, J. Phys. A, 39 (2006). Google Scholar

[10]

A. Constantin and R. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,, Fluid Dynamics Research, 40 (2008). Google Scholar

[11]

A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves,, J. Nonl. Math. Phys., 15 (2008), 58. Google Scholar

[12]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. Google Scholar

[13]

W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,", D. C. Heath and Co., (1965). Google Scholar

[14]

W. Craig, Surface water waves and tsunamis,, J. Dynam. Differential Equations, 18 (2006), 525. Google Scholar

[15]

A. Geyer, On some background flows for tsunami waves,, J. Math. Fluid Mech.., (). doi: DOI 10.1007/s00021-011-0055-0. Google Scholar

[16]

J. Hale, "Ordinary Differential Equations,", Wiley, (1969). Google Scholar

[17]

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

[18]

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

[19]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31. Google Scholar

[20]

O. Mustafa, On the uniqueness of flow in a recent tsunami model,, Applicable Analysis, (2011). doi: doi: 10.1080/00036811.2011.569499. Google Scholar

[21]

H. Segur, Waves in shallow water with emphasis on the tsunami of 2004,, in, (2007), 3. Google Scholar

[22]

R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 623. Google Scholar

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