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

Abstract

We will show that the Camassa-Holm equation possesses periodic traveling wave solutions with spikes, i.e., peaks where the first derivative is unbounded. Moreover, we will show that such a solution can be chosen to be $\rho$-periodic for arbitrarily small $\rho>0$.

This family of solutions (parametrized by $\rho$) has the important property that, for $q \in [1,3)$, $\|u_0'\|_{L^q(\mathbb{T})}$ is uniformly bounded above and below, where $u_0$ is the initial data. Using this property with $q=2$ we are able to prove that the corresponding Cauchy problem is not locally well-posed in the Sobolev space $H^1(\mathbb{T})$. Similarly, we will show ill-posedness in the corresponding $L^q$ Sobolev space, $W^{1,q}(\mathbb{T})$, for any $q\in[1,3)$.