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We show that the periodic Camassa–Holm equation $u_t-u_{xxt}+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$ possesses a global continuous semigroup of weak conservative solutions for initial data ${u|}_{t=0}$ in $H_{\mathrm{per}}^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure $\mu$ with ${\mu }_{\mathrm{ac}}=(u^2+u_x^2)dx$. The total energy is preserved by the solution.

Based on estimates for the KdV equation in analytic Gevrey classes, a spectral collocation approximation of the KdV equation is proved to converge exponentially fast.