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Global weak solutions for a shallow water equation. (English) Zbl 0930.35133

Using the method of compensated compactness, the authors prove the existence of a global weak solution to the initial value problem for the Camassa-Holm equation \(u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx}\), \(t>0\), \(x\in\mathbb{R}\), \(u(0, x)= u_0(x)\in H^1(\mathbb{R})\). This problem describes a unidirectional propagation of water waves on a free surface, and is capable of treating the interaction of peaked solutions (solitons with cusp singularities).
Reviewer: O.Titow (Berlin)

MSC:

35Q35 PDEs in connection with fluid mechanics
76B25 Solitary waves for incompressible inviscid fluids
35Q51 Soliton equations
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