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

Abstract

In this paper we consider the Holm-Staley $b$-family of equations in the Sobolev spaces $H^s(\mathbb R)$ for $s > 3/2$. 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.