The zero surface tension limit of two-dimensional water waves

David M. Ambrose, Nader Masmoudi

Research output: Contribution to journalArticle

Abstract

We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases.

Original languageEnglish (US)
Pages (from-to)1287-1315
Number of pages29
JournalCommunications on Pure and Applied Mathematics
Volume58
Issue number10
DOIs
StatePublished - Oct 2005

Fingerprint

Water waves
Water Waves
Surface Tension
Surface tension
Zero
Initial value problems
Free Surface
Initial Value Problem
Arc length
Parametrization
Existence of Solutions
Horizontal
Formulation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

The zero surface tension limit of two-dimensional water waves. / Ambrose, David M.; Masmoudi, Nader.

In: Communications on Pure and Applied Mathematics, Vol. 58, No. 10, 10.2005, p. 1287-1315.

Research output: Contribution to journalArticle

@article{93ec5b058ed64c3a966ee49973a02185,
title = "The zero surface tension limit of two-dimensional water waves",
abstract = "We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases.",
author = "Ambrose, {David M.} and Nader Masmoudi",
year = "2005",
month = "10",
doi = "10.1002/cpa.20085",
language = "English (US)",
volume = "58",
pages = "1287--1315",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "10",

}

TY - JOUR

T1 - The zero surface tension limit of two-dimensional water waves

AU - Ambrose, David M.

AU - Masmoudi, Nader

PY - 2005/10

Y1 - 2005/10

N2 - We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases.

AB - We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases.

UR - http://www.scopus.com/inward/record.url?scp=26944460000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26944460000&partnerID=8YFLogxK

U2 - 10.1002/cpa.20085

DO - 10.1002/cpa.20085

M3 - Article

VL - 58

SP - 1287

EP - 1315

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 10

ER -