Global dissipative solutions of the Camassa-Holm equation
(2007) In Analysis and Applications 5(1). p.1-27- Abstract
- This paper is devoted to the continuation of solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L-infinity space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data (u) over bar epsilon H-1(IR), and continuously depending on the initial data. The new variables resolve all singularities due to... (More)
- This paper is devoted to the continuation of solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L-infinity space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data (u) over bar epsilon H-1(IR), and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1408331
- author
- Bressan, Alberto and Constantin, Adrian LU
- organization
- publishing date
- 2007
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- non-local source, conservation law, Camassa-Holm equation, dissipative solutions
- in
- Analysis and Applications
- volume
- 5
- issue
- 1
- pages
- 1 - 27
- publisher
- World Scientific Publishing
- external identifiers
-
- wos:000252216000001
- ISSN
- 1793-6861
- DOI
- 10.1142/S0219530507000857
- language
- English
- LU publication?
- yes
- id
- 9d1731e9-34e2-4dfc-b12b-8301d793e601 (old id 1408331)
- date added to LUP
- 2016-04-01 12:11:10
- date last changed
- 2018-11-21 20:04:44
@article{9d1731e9-34e2-4dfc-b12b-8301d793e601,
abstract = {{This paper is devoted to the continuation of solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L-infinity space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data (u) over bar epsilon H-1(IR), and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.}},
author = {{Bressan, Alberto and Constantin, Adrian}},
issn = {{1793-6861}},
keywords = {{non-local source; conservation law; Camassa-Holm equation; dissipative solutions}},
language = {{eng}},
number = {{1}},
pages = {{1--27}},
publisher = {{World Scientific Publishing}},
series = {{Analysis and Applications}},
title = {{Global dissipative solutions of the Camassa-Holm equation}},
url = {{http://dx.doi.org/10.1142/S0219530507000857}},
doi = {{10.1142/S0219530507000857}},
volume = {{5}},
year = {{2007}},
}