Title: Existence time for the Camassa-Holm equation and the critical Sobolev index

Authors: Peter J. Byers

Issue: Volume 55 (2006), Issue 3, 941-954

Abstract: In this paper, we examine the break-down phenomenon for a particular type of solution to the Camassa-Holm equation---namely, a peakon-antipeakon interaction with equal magnitude. We will show, in particular, that for any $s \in (1,\frac{3}{2})$ and any $\epsilon$ greater than $0$, there exists initial data $v_0 \in H^s$ with $\| v_0 \|_{H^s}$ less than $\epsilon$ and with corresponding existence time $T$ less than $\epsilon$, for both the periodic and non-periodic cases. This implies that the initial value problem for the Camassa-Holm equation is not locally well-posed in $H^s$ for any $s$ less than $\frac{3}{2}$. Since it is already known that this initial value problem is locally well-posed in every Sobolev space $H^s$ with $s$ greater than $\frac{3}{2}$, we conclude that $s_0 = \frac{3}{2}$ is the critical Sobolev index for the Camassa-Holm equation.