Periodic conservative solutions of the Camassa–Holm equation

Holden, Helge; Raynaud, Xavier

doi:10.5802/aif.2375

Periodic conservative solutions of the Camassa–Holm equation
[Solutions périodiques conservatives de l’équation de Camassa–Holm]

Holden, Helge ¹ ; Raynaud, Xavier 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, Tome 58 (2008) no. 3, pp. 945-988.

Résumé
Abstract

Nous montrons que l’équation de Camassa–Holm périodique $u_{t} - u_{x x t} + 3 u u_{x} - 2 u x u_{x x} - u u_{x x x} = 0$ possède un semi-groupe continu de solutions globales pour des conditions initiales ${u |}_{t = 0}$ dans $H_{per}^{1}$ . 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 $μ$ qui satisfait $μ_{ac} = (u^{2} + u_{x}^{2}) d x$ . L’énergie totale est préservée par la solution.

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.

MR Zbl

DOI : 10.5802/aif.2375

Classification : 65M06, 65M12, 35B10, 35Q53
Keywords: Camassa–Holm equation, periodic solution
Mot clés : équation de Camassa–Holm, solutions périodiques

Affiliations des auteurs :

Holden, Helge ¹ ; Raynaud, Xavier ²

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     title = {Periodic conservative solutions of the {Camassa{\textendash}Holm} equation},
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Holden, Helge; Raynaud, Xavier. Periodic conservative solutions of the Camassa–Holm equation. Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 945-988. doi : 10.5802/aif.2375. https://www.numdam.org/articles/10.5802/aif.2375/

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