Overhanging and touching waves in constant vorticity flows

Vera Hur; Miles Wheeler

doi:10.1016/j.jde.2022.08.012

Overhanging and touching waves in constant vorticity flows

Vera Hur, Miles Wheeler

Department of Mathematical Sciences

University of Illinois at Urbana-Champaign

Research output: Contribution to journal › Article › peer-review

7 Citations (SciVal)

102 Downloads (Pure)

Abstract

We show the existence of periodic traveling waves at the free surface of a two dimensional,infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper’s celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity

Original language	English
Pages (from-to)	572-590
Number of pages	19
Journal	Journal of Differential Equations
Volume	338
Early online date	27 Aug 2022
DOIs	https://doi.org/10.1016/j.jde.2022.08.012
Publication status	Published - 25 Nov 2022

Bibliographical note

Funding Information:
The work of VMH was supported by the NSF through the award DMS-2009981 . The authors are also indebted to the anonymous referee for their helpful comments, and especially for identifying and resolving an error in an earlier version of Lemma 5 .

Keywords

Constant vorticity
Crapper
Free surface waves
Overhanging
Periodic traveling
Touching

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jde.2022.08.012Licence: CC BY

touching_preprintSubmitted manuscript, 501 KB

Cite this

@article{0631f460d2134fbd9ea8807a79cefabb,

title = "Overhanging and touching waves in constant vorticity flows",

abstract = "We show the existence of periodic traveling waves at the free surface of a two dimensional,infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper{\textquoteright}s celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity",

keywords = "Constant vorticity, Crapper, Free surface waves, Overhanging, Periodic traveling, Touching",

author = "Vera Hur and Miles Wheeler",

note = "Funding Information: The work of VMH was supported by the NSF through the award DMS-2009981 . The authors are also indebted to the anonymous referee for their helpful comments, and especially for identifying and resolving an error in an earlier version of Lemma 5 . ",

year = "2022",

month = nov,

day = "25",

doi = "10.1016/j.jde.2022.08.012",

language = "English",

volume = "338",

pages = "572--590",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Elsevier Academic Press Inc",

}

TY - JOUR

T1 - Overhanging and touching waves in constant vorticity flows

AU - Hur, Vera

AU - Wheeler, Miles

N1 - Funding Information: The work of VMH was supported by the NSF through the award DMS-2009981 . The authors are also indebted to the anonymous referee for their helpful comments, and especially for identifying and resolving an error in an earlier version of Lemma 5 .

PY - 2022/11/25

Y1 - 2022/11/25

N2 - We show the existence of periodic traveling waves at the free surface of a two dimensional,infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper’s celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity

AB - We show the existence of periodic traveling waves at the free surface of a two dimensional,infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper’s celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity

KW - Constant vorticity

KW - Crapper

KW - Free surface waves

KW - Overhanging

KW - Periodic traveling

KW - Touching

UR - http://www.scopus.com/inward/record.url?scp=85136480862&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2022.08.012

DO - 10.1016/j.jde.2022.08.012

M3 - Article

SN - 0022-0396

VL - 338

SP - 572

EP - 590

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

Overhanging and touching waves in constant vorticity flows

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this