Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves
(2013) In Journal of Differential Equations 254(3). p.1006-1096- Abstract
- In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy, which is given in dimensionless coordinates by epsilon(eta, phi) = integral(R2){1/2 integral(1+eta)(0) (phi(2)(x) + phi(2)(y) + phi(2)(z))dy +1/2 eta(2) + beta[root 1 + eta(2)(x) + eta(2)(z) - 1]}dxdz, subject to the constraint that the momentum I(eta, phi) = integral(R2)eta(x)phi vertical bar(y=1+eta)dzdz is fixed; here {(x, y, z): x, z is an element of R, y is an element... (More)
- In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy, which is given in dimensionless coordinates by epsilon(eta, phi) = integral(R2){1/2 integral(1+eta)(0) (phi(2)(x) + phi(2)(y) + phi(2)(z))dy +1/2 eta(2) + beta[root 1 + eta(2)(x) + eta(2)(z) - 1]}dxdz, subject to the constraint that the momentum I(eta, phi) = integral(R2)eta(x)phi vertical bar(y=1+eta)dzdz is fixed; here {(x, y, z): x, z is an element of R, y is an element of (0, 1 + eta(x, z))} is the fluid domain, phi is the velocity potential and beta > 1/3 is the Bond number. These functionals are studied locally for eta in a neighbourhood of the origin in H-3(R-2). We prove the existence of a minimiser of epsilon subject to the constraint I = 2 mu, where 0 < mu << 1. The existence of a small-amplitude solitary wave is thus assured, and since epsilon and I are both conserved quantities a standard argument may be used to establish the stability of the set D-mu of minimisers as a whole. 'Stability is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves. We show that solutions to the evolutionary problem starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence; they may however explode in finite time due to higher-order derivatives becoming unbounded. (C) 2012 Elsevier Inc. All rights reserved. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3400668
- author
- Buffoni, B. ; Groves, M. D. ; Sun, S. M. and Wahlén, Erik LU
- organization
- publishing date
- 2013
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Journal of Differential Equations
- volume
- 254
- issue
- 3
- pages
- 1006 - 1096
- publisher
- Elsevier
- external identifiers
-
- wos:000312574500003
- scopus:84870372052
- ISSN
- 0022-0396
- DOI
- 10.1016/j.jde.2012.10.007
- language
- English
- LU publication?
- yes
- id
- 32e9f494-8e57-4d53-9023-5c04afef671b (old id 3400668)
- date added to LUP
- 2016-04-01 11:15:37
- date last changed
- 2022-04-28 08:28:58
@article{32e9f494-8e57-4d53-9023-5c04afef671b,
abstract = {{In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy, which is given in dimensionless coordinates by epsilon(eta, phi) = integral(R2){1/2 integral(1+eta)(0) (phi(2)(x) + phi(2)(y) + phi(2)(z))dy +1/2 eta(2) + beta[root 1 + eta(2)(x) + eta(2)(z) - 1]}dxdz, subject to the constraint that the momentum I(eta, phi) = integral(R2)eta(x)phi vertical bar(y=1+eta)dzdz is fixed; here {(x, y, z): x, z is an element of R, y is an element of (0, 1 + eta(x, z))} is the fluid domain, phi is the velocity potential and beta > 1/3 is the Bond number. These functionals are studied locally for eta in a neighbourhood of the origin in H-3(R-2). We prove the existence of a minimiser of epsilon subject to the constraint I = 2 mu, where 0 < mu << 1. The existence of a small-amplitude solitary wave is thus assured, and since epsilon and I are both conserved quantities a standard argument may be used to establish the stability of the set D-mu of minimisers as a whole. 'Stability is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves. We show that solutions to the evolutionary problem starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence; they may however explode in finite time due to higher-order derivatives becoming unbounded. (C) 2012 Elsevier Inc. All rights reserved.}},
author = {{Buffoni, B. and Groves, M. D. and Sun, S. M. and Wahlén, Erik}},
issn = {{0022-0396}},
language = {{eng}},
number = {{3}},
pages = {{1006--1096}},
publisher = {{Elsevier}},
series = {{Journal of Differential Equations}},
title = {{Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves}},
url = {{http://dx.doi.org/10.1016/j.jde.2012.10.007}},
doi = {{10.1016/j.jde.2012.10.007}},
volume = {{254}},
year = {{2013}},
}