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 $H^s(\mathbb{T})$, $s>3/2$.