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Global Conservative Solutions Of The Camassa–Holm Equation—A Lagrangian Point Of View

H. Holden, X. Raynaud
Published 2007 · Mathematics

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We show that the Camassa–Holm equation ut − uxxt + 3uux − 2uxuxx − uuxxx = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u|t=0 in H1. 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 . The total energy is preserved by the solution.
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