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Global dissipative solutions of the Camassa-Holm equation

Bressan, Alberto and Constantin, Adrian LU (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:
author
organization
publishing date
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
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
2009-05-29 12:52:49
date last changed
2016-04-15 19:53:39
@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},
  keyword      = {non-local source,conservation law,Camassa-Holm equation,dissipative solutions},
  language     = {eng},
  number       = {1},
  pages        = {1--27},
  publisher    = {World Scientific},
  series       = {Analysis and Applications},
  title        = {Global dissipative solutions of the Camassa-Holm equation},
  url          = {http://dx.doi.org/10.1142/S0219530507000857},
  volume       = {5},
  year         = {2007},
}