On the well-posedness of the hyperelastic rod equation

TL;DR

This paper investigates the hyperelastic rod equation in Sobolev spaces, demonstrating that its solution map lacks local uniform continuity, which has implications for the equation's well-posedness and stability.

Contribution

It establishes the non-uniform continuity of the solution map for the hyperelastic rod equation using a geometric approach, extending results to both real line and periodic cases.

Findings

Solution map is nowhere locally uniformly continuous

Results apply to both Sobolev spaces on R and T

Highlights challenges in the well-posedness of the equation

Abstract

In this paper we consider the hyperelastic rod equation on the Sobolev spaces $H^s(\R)$, $s > 3/2$. 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 $H^s(\mathbb T)$, $s > 3/2$.