Ill-posedness in the critical Sobolev space for the hyperelastic rod equation
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 H3/2(ℝ). We prove ill-posedness at this endpoint, complementing the local well-posedness theory in Hs(ℝ) for s>3/2 and the recently established norm-inflation result for 1<s<3/2. The nature of the critical ill-posedness depends on the parameter γ. For 0<γ≤3, we construct arbitrarily small smooth initial data whose corresponding solutions develop finite-time blow-up, with the maximal lifespan tending to zero with the size of the initial data. For γ<0 or γ>3, we construct solutions with arbitrarily small H3/2 initial data whose H3/2 norm becomes arbitrarily large in arbitrarily short time.
Related
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BibTeX
@misc{geba2026illposedness,
title = {{Ill-posedness in the critical Sobolev space for the hyperelastic rod equation}},
author = {Dan-Andrei Geba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00594},
eprint = {2610.00594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00594},
}
RIS
TY - UNPB
TI - Ill-posedness in the critical Sobolev space for the hyperelastic rod equation
AU - Dan-Andrei Geba
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00594
UR - https://arxiv.org/abs/2610.00594
AB - 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(ℝ). We prove ill-posedness at this endpoint, complementing the local well-posedness theory in H^s(ℝ) for s>3/2 and the recently established norm-inflation result for 1<s<3/2. The nature of the critical ill-posedness depends on the parameter γ. For 0<γ≤3, we construct arbitrarily small smooth initial data whose corresponding solutions develop finite-time blow-up, with the maximal lifespan tending to zero with the size of the initial data. For γ<0 or γ>3, we construct solutions with arbitrarily small H^3/2 initial data whose H^3/2 norm becomes arbitrarily large in arbitrarily short time.
ER -