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Asymptotics of two-dimensional hydroelastic waves: The zero mass, zero bending limit
Journal article   Open access   Peer reviewed

Asymptotics of two-dimensional hydroelastic waves: The zero mass, zero bending limit

David M Ambrose and Shunlian Liu
Journal of differential equations, v 424, pp 381-420
10 Jan 2025
DOI: https://doi.org/10.1016/j.jde.2024.12.051
Featured in Collection :   Research Supported by Drexel Libraries' OA Programs
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https://doi.org/10.1016/j.jde.2024.12.051View
Published, Version of Record (VoR)Open Access via Drexel Libraries Read and Publish Program 2024CC BY V4.0,  Open

Abstract

Partial Differential Equations
We consider two-dimensional hydroelastic waves, in which a free fluid surface separates two fluids of infinite vertical extent. Elastic effects are accounted for at the interface, with a parameter measuring the elastic bending force and another parameter measuring the mass of the elastic sheet. In prior work, the authors have demonstrated well-posedness of this initial value problem in Sobolev spaces. We now take the limit as these two parameters vanish. Since the size of the time interval of existence given by this prior theory vanishes as the mass and bending parameters go to zero, we now establish estimates which are uniform with respect to these parameters. We may then make an additional estimate which demonstrates that the solutions form a Cauchy sequence as the parameters go to zero, so that the limit may be taken. This demonstrates that the vortex sheet with surface tension is the zero mass, zero bending limit of hydroelastic waves in two spatial dimensions.

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Mathematics
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