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# Global weak solutions to the Camassa-Holm equation

• The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition $u_0(x)$ has end states $u_{\pm}$, which has much weaker constraints than that $u_0(x) \in H^1(\mathbb R)$ discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption $u_- < u_+$.
Mathematics Subject Classification: Primary: 35D05, 35L60.

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