# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [1] 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 [2] 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 [3] Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103 [18] 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 [19] Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795 [20] Gerhard Rein. Galactic dynamics in MOND---Existence of equilibria with finite mass and compact support. Kinetic & Related Models, 2015, 8 (2) : 381-394. doi: 10.3934/krm.2015.8.381

2019 Impact Factor: 1.105