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Periodic conservative solutions of the Camassa-Holm equation. (English) Zbl 1158.35079
The authors study the periodic Camassa-Holm equation
\[ u_t-u_{xxt} +3u u_x-2u_x u_{xx} - u u_{xxx}=0. \] It is proved that a global continuous semigroup of weak conservative solutions for initial data \(u| _{t=0}\) in \(H_{\text{per}}'\) exists. The result is obtained by introducing 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 \(\mu_{ac} =(u^2 +u_x^2)dx\). The total energy is shown to be preserved by the solution.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B10 Periodic solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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