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A few remarks on the Camassa-Holm equation. (English) Zbl 1161.35329

Summary: In the present paper, we use some standard a priori estimates for linear transport equations to prove the existence and uniqueness of solutions for the Camassa-Holm equation with minimal regularity assumptions on the initial data. We also derive some explosion criteria and a sharp estimate from below for the existence time. We finally address the question of global existence for certain initial data. This yields, among other things, a different proof for the existence and uniqueness of Constantin and Molinet’s global weak solutions

MSC:

35B65 Smoothness and regularity of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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