Periodic conservative solutions of the Camassa–Holm equation

[Solutions périodiques conservatives de l’équation de Camassa–Holm]

Holden, Helge ; Raynaud, Xavier

Annales de l'Institut Fourier, Tome 58 (2008), p. 945-988 / Harvested from Numdam

Résumé

Nous montrons que l’équation de Camassa–Holm périodique $u_t - u_{xxt} + 3uu_x - 2ux u_{xx} - uu_{xxx} = 0$ possède un semi-groupe continu de solutions globales pour des conditions initiales $u|_{t=0}$ dans $H_{\rm per}^1$. Le résultat est obtenu en utilisant un changement de variable où l’équation est réécrite en variables lagrangiennes. Pour décrire les solutions, il est nécessaire d’introduire la densité d’énergie donnée par la mesure de Radon positive $\mu $ qui satisfait $\mu _{\rm ac}=(u^2+u_x^2)dx$. L’énergie totale est préservée par la solution.

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.

Détail

Publié le : 2008-01-01

MR 2427516 | Zbl 1158.35079

DOI : https://doi.org/10.5802/aif.2375
Classification:  65M06,  65M12,  35B10,  35Q53
Mots clés: équation de Camassa–Holm, solutions périodiques

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