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On the integrability of KdV hierarchy with self-consistent sources
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
Anna Geyer 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.
Keywords: tsunami background flow, uniqueness and compact support of solutions., Nonlinear water waves with vorticity.
Mathematics Subject Classification: Primary: 34A34, 34A12; Secondary: 35Q31, 35J6.
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
G. K. Batchelor, "An Introduction to Fluid Dynamics,", Cambridge University Press, (1967). Google Scholar
E. Bryant, "Tsunami: The Underrated Hazard,", Springer Praxis Books, (2008). Google Scholar
A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. Google Scholar
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
A. Constantin, On the relevance of soliton theory to tsunami modelling,, Wave Motion, 46 (2009), 420. Google Scholar
A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states,, Arch. Ration. Mech. Anal., 200 (2011), 239. Google Scholar
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 172 (2010). Google Scholar
A. Constantin and D. Henry, Solitons and tsunamis,, Z. Naturforsch., 64a (2009), 65. Google Scholar
A. Constantin and R. S. Johnson, Modelling tsunamis,, J. Phys. A, 39 (2006). Google Scholar
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
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
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. Google Scholar
W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,", D. C. Heath and Co., (1965). Google Scholar
W. Craig, Surface water waves and tsunamis,, J. Dynam. Differential Equations, 18 (2006), 525. Google Scholar
A. Geyer, On some background flows for tsunami waves,, J. Math. Fluid Mech.., (). doi: DOI 10.1007/s00021-011-0055-0. Google Scholar
J. Hale, "Ordinary Differential Equations,", Wiley, (1969). Google Scholar
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
R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,", Cambridge University Press, (1997). Google Scholar
M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31. Google Scholar
O. Mustafa, On the uniqueness of flow in a recent tsunami model,, Applicable Analysis, (2011). doi: doi: 10.1080/00036811.2011.569499. Google Scholar
H. Segur, Waves in shallow water with emphasis on the tsunami of 2004,, in, (2007), 3. Google Scholar
R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 623. Google Scholar
Walter A. Strauss. Vorticity jumps in steady water waves. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101
Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191
Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607
Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397
Mats Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 483-491. doi: 10.3934/dcds.2007.19.483
Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109
Adrian Constantin. On the propagation of tsunami waves, with emphasis on the tsunami of 2004. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 525-537. doi: 10.3934/dcdsb.2009.12.525
Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475
Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114
Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045
Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5521-5541. doi: 10.3934/dcds.2019225
Raphael Stuhlmeier. Effects of shear flow on KdV balance - applications to tsunami. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1549-1561. doi: 10.3934/cpaa.2012.11.1549
Bogdan-Vasile Matioc. A characterization of the symmetric steady water waves in terms of the underlying flow. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3125-3133. doi: 10.3934/dcds.2014.34.3125
Fabio Punzo. Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 657-670. doi: 10.3934/dcdss.2012.5.657
R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497
Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523
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Colm Connaughton, John R. Ockendon. Interactions of point vortices in the Zabusky-McWilliams model with a background flow. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1795-1807. doi: 10.3934/dcdsb.2012.17.1795