Holden, Helge; Raynaud, Xavier
Periodic conservative solutions of the Camassa–Holm equation  [ Solutions périodiques conservatives de l’équation de Camassa–Holm ]
Annales de l'institut Fourier, Tome 58 (2008) no. 3 , p. 945-988
MR 2427516 | Zbl 1158.35079
doi : 10.5802/aif.2375
URL stable : http://www.numdam.org/item?id=AIF_2008__58_3_945_0

Classification:  65M06,  65M12,  35B10,  35Q53
Mots clés: équation de Camassa–Holm, solutions périodiques
Nous montrons que l’équation de Camassa–Holm périodique u t -u xxt +3uu x -2uxu xx -uu xxx =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 )dx. L’énergie totale est préservée par la solution.
We show that the periodic Camassa–Holm equation u t -u xxt +3uu x -2u x u xx -uu xxx =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 )dx. The total energy is preserved by the solution.

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