Global weak solutions to the Camassa-Holm equation
The authors establish global weak solutions of the Camassa--Holm equation, \[ u_t - u_{txx} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}, \] for initial data \(u_0(x)\) with nontrivial end states, \(\lim_{x \to \pm \infty} u_0(x) = u_{\pm}\). It is assumed that \(u_{-} < u_+\), whence initial data \(u_0 \in H^1(\mathbb R)\) are excluded. Instead, it is required that \(u_0 - \phi_0 \in H^1(\mathbb R)\), for a function \[ \phi_{0}(x) := \frac{u_{+} + u_{-}}{2} + \frac{u_{+} - u_{-}}{2} K_q \int_0^{\sigma x} \left(1+y^2 \right)^{-q}\, dy, \qquad \sigma > 0, \] such that for each \(q > 1/2\), we have \(K_q \int_0^\infty (1+y^2)^{-q}\,dy = 1\). It is proved that there exists a global solution \(u(t,x)\), such that if \(\phi(t,x)\) with \(\phi(0,x) = \phi_0(x)\) satisfies \[ \phi_t + 3\phi \phi_x = 0, \] then \(\| u-\phi \| _{H^1(\mathbb R)} \leq C \left( \| u_0 - \phi_0 \| _{H^1(\mathbb R)} +1 \right)\), for all \(t > 0\). Furthermore, \(u\) approaches \(\phi\) pointwise as \(t \to \infty\). The paper can be considered an extension of the investigation by \textit{Z. Xin} and \textit{P. Zhang} [Commun. Pure Appl. Math. 53, No.~11, 1411--1433 (2000; Zbl 1048.35092)], and some of the methods are based on ideas developed in the same paper. For related results, see also the recent article by \textit{A. Bressan} and \textit{A. Constantin} [Arch. Ration. Mech. Anal. 183, No.~2, 215--239 (2007; Zbl 1105.76013)].
scientific article; zbMATH DE number 5304869
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