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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