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arXiv · 2610.00594

Ill-posedness in the critical Sobolev space for the hyperelastic rod equation

Dan-Andrei Geba
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Abstract

This article is the third in a series investigating ill-posedness in critical Sobolev spaces for Camassa-Holm-type equations. Following results for the $b$-Novikov and Fokas-Olver-Rosenau-Qiao equations, we study the hyperelastic rod equation at the critical Sobolev regularity $H^{3/2}(\mathbb R)$. We prove ill-posedness at this endpoint, complementing the local well-posedness theory in $H^s(\mathbb R)$ for $s>3/2$ and the recently established norm-inflation result for $1 3$, we construct solutions with arbitrarily small $H^{3/2}$ initial data whose $H^{3/2}$ norm becomes arbitrarily large in arbitrarily short time.

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Dan-Andrei Geba. 2026-09-30. Ill-posedness in the critical Sobolev space for the hyperelastic rod equation. https://arxiv.org/abs/2610.00594

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