On the well-posedness of the Holm-Staley b-family of equations

In this paper we consider the Holm-Staley $b$-family of equations in the Sobolev spaces $H^s(\mathbb{R})$ for $s>3\mathrm{/}2$. Using a geometric approach we show that, for any value of the parameter $b$, the corresponding solution map,$u(0)\mapsto u(T)$, is nowhere locally uniformly continuous.