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

Abstract

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We show that the periodic Camassa–Holm equation u t - u x x t + 3 u u x - 2 u x u x x - u u x x x = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u | t = 0 in H 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 μ with μ ac = ( u 2 + u x 2 ) d x . The total energy is preserved by the solution.

How to cite

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Holden, 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 -

References

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