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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 ${u}_{t}-{u}_{xxt}+3u{u}_{x}=2{u}_{x}{u}_{xx}+u{u}_{xxx}$, $t>0$, $x\in ℝ$, $u\left(0,x\right)={u}_{0}\left(x\right)\in {H}^{1}\left(ℝ\right)$. 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 (inviscid fluids) 35Q51 Soliton-like equations