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Well-posedness of two-dimensional hydroelastic waves

Published online by Cambridge University Press:  16 March 2017

David M. Ambrose
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA (ambrose@math.drexel.edu)
Michael Siegel
Affiliation:
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA (misieg@njit.edu)

Extract

A well-posedness theory for the initial-value problem for hydroelastic waves in two spatial dimensions is presented. This problem, which arises in numerous applications, describes the evolution of a thin elastic membrane in a two-dimensional (2D) potential flow. We use a model for the elastic sheet that accounts for bending stresses and membrane tension, but which neglects the mass of the membrane. The analysis is based on a vortex sheet formulation and, following earlier analyses and numerical computations in 2D interfacial flow with surface tension, we use an angle–arclength representation of the problem. We prove short-time well-posedness in Sobolev spaces. The proof is based on energy estimates, and the main challenge is to find a definition of the energy and estimates on high-order non-local terms so that an a priori bound can be obtained.

MSC classification

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2017 

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