On the well-posedness of the hyperelastic rod equation

Abstract

In this paper we consider the hyperelastic rod equation on the Sobolev spaces \(H^s({\mathbb {R}})\), \(s > 3/2\). Using a geometric approach we show that for any \(T > 0\) the corresponding solution map, \(u(0) \mapsto u(T)\), is nowhere locally uniformly continuous. The method applies also to the periodic case \(H^s({\mathbb {T}})\), \(s > 3/2\).