# Periodic conservative solutions of the Camassa–Holm equation

Helge Holden^{[1]}; Xavier Raynaud^{[2]}

- [1] Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway) University of Oslo Centre of Mathematics for Applications P.O. Box 1053, Blindern 0316 Oslo (Norway)
- [2] Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway)

Annales de l’institut Fourier (2008)

- Volume: 58, Issue: 3, page 945-988
- ISSN: 0373-0956

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topHolden, Helge, and Raynaud, Xavier. "Periodic conservative solutions of the Camassa–Holm equation." Annales de l’institut Fourier 58.3 (2008): 945-988. <http://eudml.org/doc/10340>.

@article{Holden2008,

abstract = {We show that the periodic Camassa–Holm equation $u_t-u_\{xxt\}+3uu_x-2u_xu_\{xx\}-uu_\{xxx\}=0$ possesses a global continuous semigroup of weak conservative solutions for initial data $u|_\{t=0\}$ in $H_\{\rm per\}^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure $\mu $ with $\mu _\{\rm ac\}=(u^2+u_x^2)dx$. The total energy is preserved by the solution.},

affiliation = {Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway) University of Oslo Centre of Mathematics for Applications P.O. Box 1053, Blindern 0316 Oslo (Norway); Norwegian University of Science and Technology Department of Mathematical Sciences 7491 Trondheim (Norway)},

author = {Holden, Helge, Raynaud, Xavier},

journal = {Annales de l’institut Fourier},

keywords = {Camassa–Holm equation; periodic solution; periodic conservative solution; Lagrangian coordinate},

language = {eng},

number = {3},

pages = {945-988},

publisher = {Association des Annales de l’institut Fourier},

title = {Periodic conservative solutions of the Camassa–Holm equation},

url = {http://eudml.org/doc/10340},

volume = {58},

year = {2008},

}

TY - JOUR

AU - Holden, Helge

AU - Raynaud, Xavier

TI - Periodic conservative solutions of the Camassa–Holm equation

JO - Annales de l’institut Fourier

PY - 2008

PB - Association des Annales de l’institut Fourier

VL - 58

IS - 3

SP - 945

EP - 988

AB - We show that the periodic Camassa–Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ possesses a global continuous semigroup of weak conservative solutions for initial data $u|_{t=0}$ in $H_{\rm per}^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure $\mu $ with $\mu _{\rm ac}=(u^2+u_x^2)dx$. The total energy is preserved by the solution.

LA - eng

KW - Camassa–Holm equation; periodic solution; periodic conservative solution; Lagrangian coordinate

UR - http://eudml.org/doc/10340

ER -

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