Spiked traveling waves and ill-posedness for the Camassa-Holm equation on the circle (Q1766869)

The following partial differential equation is considered: \[ \partial _t u - \partial _t \partial _x^2 u + \frac{3}{2}\partial _x (u^2) - \frac{1}{2}\partial _x^3 (u^2) + \frac{1}{2}\partial _x (\partial _x (u)^2) = 0. \] This equation is formally equivalent to the equation \[ \partial _t u - \partial _x^2 \partial _t u + 3u\partial _x u - 2\partial _x u\partial _x^2 u - u\partial _x^3 u = 0, \] which now is generally called the Camassa-Holm equation and was derived in different ways by and Fokas and Fuchssteiner and by Camassa and Holm. The corresponding periodic initial value problem is also studied, namely \[ \begin{cases} \partial _t u - \partial _t \partial _x^2 u + \frac{3}{2}\partial _x (u^2) - \frac{1}{2}\partial _x^3 (u^2) + \frac{1}{2}\partial _x (\partial _x (u)^2) = 0, \\ u(x,0) = u_0 (x), \end{cases} \] where \(t \in {\mathcal R}\) and \(x \in {\mathcal T}.\) This initial value problem is locally well-posed in a Banach space \(E\) if for every \(r > 0\) there exists \(T > 0\) such that (i) for each \( u(x,0) = u_0 (x) \in B(0,r) = \{\varphi \in E \mid \| \varphi \|_E \leq r\}\), there exists a unique solution \(u = u(x,t) \in C([ - T,T]:E)\) of the initial value problem; (ii) the map from \(B(0,r)\) into \(C([- T,T]:E)\) given by \(u_0 \mapsto u\) is uniformly continuous. The definition implies, in particular, that for each \(t \in [- T,T],\;u_0 \mapsto u(t)\) is a uniformly continuous map from \(B(0,r)\) into \(E,\) where \(u(t)\) is understood to mean \(u(- ,t)\) . The main result in the paper is that the periodic Camassa-Holm equation is not well-posed in \(H^1. \)