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

Shunlian Liu,David M Ambrose

doi:10.3934/dcds.2019129

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.

Highlights

In studying the motion of free surfaces in fluid dynamics, many approximate models have been introduced in order to make problems more tractable
The second author, Bona, and Nicholls have previously presented strong evidence that quadratic and cubic truncated series models of gravity water waves are ill-posed [3]; for the quadratic case, the second author and Siegel are working on a full proof of ill-posedness [7]
That the initial value problems for these systems are ill-posed is in one sense surprising, since the full equations of motion for gravity water waves are well-posed

Summary

Introduction

In studying the motion of free surfaces in fluid dynamics, many approximate models have been introduced in order to make problems more tractable. While the truncated series models for gravity water waves appear to have ill-posed initial value problems, dispersion is well-known to offer regularizing effects [12], [21]. In the case p = 2, the system (1.2) has the correct dispersion relation for the presence of surface tension, and the system may be regarded as a quadratic truncated series model for pure capillary waves Note that in this case, p = 2, the question of well-posedness or illposedness is at present unresolved. We prove existence and uniqueness of solutions for the initial value problem for (1.2) for p ≥ 3, which includes the hydroelastic case We do this by using paradifferential calculus as developed by Bony [10] (see [20], [25]).

Preliminaries

Existence

Conclusion