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
$\rho>0$.
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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 $q\in[1,3)$.