Well-posedness of two-dimensional hydroelastic waves

David M. Ambrose, Michael Siegel

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)529-570
Number of pages42
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume147
Issue number3
DOIs
StatePublished - Jun 1 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • energy method
  • hydroelastic waves
  • vortex sheet
  • well-posedness

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