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Helge Holden; Xavier Raynaud
Periodic conservative solutions of the Camassa–Holm equation
(Solutions périodiques conservatives de l’équation de Camassa–Holm)
Annales de l'institut Fourier, 58 no. 3 (2008), p. 945-988, doi: 10.5802/aif.2375
Article PDF | Reviews MR 2427516 | Zbl 1158.35079
Class. Math.: 65M06, 65M12, 35B10, 35Q53
Keywords: Camassa–Holm equation, periodic solution

Résumé - 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.

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