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Zbl 1136.35080
Holden, Helge; Raynaud, Xavier
Global conservative solutions of the Camassa-Holm equation -- a Lagrangian point of view. (English)
[J] Commun. Partial Differ. Equations 32, No. 10, 1511-1549 (2007). ISSN 0360-5302; ISSN 1532-4133/e

The authors show that the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ possesses a global continuous semigroup of weak conservative solutions provided the initial data $u\mid_{t=0}$ belongs to $H^1$. The result is obtained by using Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure $\mu$ with $\mu_{ac}=(u^2+u^2_x)$. The total energy is preserved by the solution.
[Marius Ghergu (Dublin)]

MSC 2000:: *35Q53 KdV-like equations

Keywords: Camassa-Holm equation; conservative solutions; Lagrangian coordinates

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