On the Stability of Solitary Water Waves with a Point Vortex
(2020) In Communications on Pure and Applied Mathematics 73(12). p.2634-2684- Abstract
This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss (GSS), but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the... (More)
This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss (GSS), but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state-dependent. As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized Korteweg–de Vries (KdV) and Benjamin-Ono equations. The stability or instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss, who used a modification of the GSS method. We provide a new, more direct proof of these results, as a straightforward consequence of our abstract theory. At the same time, we allow fractional dispersion and obtain a new instability result for fractional KdV.
(Less)
- author
- Varholm, Kristoffer LU ; Wahlén, Erik LU and Walsh, Samuel
- organization
- publishing date
- 2020-12
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Communications on Pure and Applied Mathematics
- volume
- 73
- issue
- 12
- pages
- 51 pages
- publisher
- John Wiley & Sons Inc.
- external identifiers
-
- scopus:85082967082
- ISSN
- 0010-3640
- DOI
- 10.1002/cpa.21891
- project
- Nonlinear water waves and nonlocal model equations
- Nonlinear Water Waves
- language
- English
- LU publication?
- yes
- id
- 82ca718c-8d05-4a04-be79-cc5cd5e980a3
- date added to LUP
- 2020-05-07 16:50:21
- date last changed
- 2022-05-12 02:10:15
@article{82ca718c-8d05-4a04-be79-cc5cd5e980a3,
abstract = {{<p>This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss (GSS), but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state-dependent. As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized Korteweg–de Vries (KdV) and Benjamin-Ono equations. The stability or instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss, who used a modification of the GSS method. We provide a new, more direct proof of these results, as a straightforward consequence of our abstract theory. At the same time, we allow fractional dispersion and obtain a new instability result for fractional KdV.</p>}},
author = {{Varholm, Kristoffer and Wahlén, Erik and Walsh, Samuel}},
issn = {{0010-3640}},
language = {{eng}},
number = {{12}},
pages = {{2634--2684}},
publisher = {{John Wiley & Sons Inc.}},
series = {{Communications on Pure and Applied Mathematics}},
title = {{On the Stability of Solitary Water Waves with a Point Vortex}},
url = {{http://dx.doi.org/10.1002/cpa.21891}},
doi = {{10.1002/cpa.21891}},
volume = {{73}},
year = {{2020}},
}