Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves

Shunlian Liu; David M. Ambrose

doi:10.3934/dcds.2019129

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2019, Volume 39, Issue 6: 3123-3147. Doi: 10.3934/dcds.2019129

This issue Previous Article Classification of traveling waves for a quadratic Szegő equation Next Article On substitution tilings and Delone sets without finite local complexity

Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves

Shunlian Liu^1, and
David M. Ambrose^2, ,

1.
School of Science, Hunan University of Technology, Zhuzhou, Hunan 412007, China
2.
Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

^* Corresponding author: David M. Ambrose
^* Corresponding author: David M. Ambrose

Received: March 31, 2018

Revised: July 31, 2018

Published: February 2019

The second author is grateful for support from the National Science Foundation through grant DMS-1515849.

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Abstract

Truncated series models of gravity water waves are popular for use in simulation. Recent work has shown that these models need not inherit the well-posedness properties of the full equations of motion (the irrotational, incompressible Euler equations). We show that if one adds a sufficiently strong dispersive term to a quadratic truncated series model, the system then has a well-posed initial value problem. Such dispersion can be relevant in certain physical contexts, such as in the case of a bending force present at the free surface, as in a hydroelastic sheet.

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Mathematics Subject Classification: Primary: 76B15; Secondary: 34A12, 76B07, 35S50.

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