Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Computing time-periodic solutions of a model for the vortex sheet with surface tension


Authors: David M. Ambrose, Mark Kondrla, Jr. and Michael Valle
Journal: Quart. Appl. Math. 73 (2015), 317-329
MSC (2010): Primary 35B10; Secondary 37M20, 76B45
DOI: https://doi.org/10.1090/S0033-569X-2015-01364-8
Published electronically: March 20, 2015
MathSciNet review: 3357496
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Abstract: We compute time-periodic solutions of a simple model for the vortex sheet with surface tension. The model has the same dispersion relation as the full system of evolution equations, and it also has the same destabilizing nonlinearity (if the surface tension parameter were to be set to zero, then this nonlinearity would cause an analogue of the Kelvin-Helmholtz instability). The numerical method uses a gradient descent algorithm to minimize a functional which measures whether a solution of the system is time periodic. We find continua of genuinely time-periodic solutions bifurcating from equilibrium.


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Additional Information

David M. Ambrose
Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email: ambrose@math.drexel.edu

Mark Kondrla, Jr.
Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email: mKondrla415@gmail.com

Michael Valle
Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email: mikau16@hotmail.com

DOI: https://doi.org/10.1090/S0033-569X-2015-01364-8
Keywords: vortex sheet, surface tension, time-periodic, gradient descent
Received by editor(s): March 30, 2013
Published electronically: March 20, 2015
Additional Notes: The authors gratefully acknowledge support from the National Science Foundation through NSF grants DMS-1008387 and DMS-1016267. We also gratefully acknowledge support in the form of Research Co-Op Funding from the Drexel University Office of the Provost and the Steinbright Career Development Office.
Article copyright: © Copyright 2015 Brown University


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