Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation
Abstract
We study the Hunter--Saxton equation on the real line and its global dissipative solution in the energy space $ L^\infty(\mathbb R)\cap \dot H^1(\mathbb R). $ We prove strong ill-posedness through instantaneous failure of Sobolev regularity at and below the Lipschitz threshold. More precisely, for every s∈(1,nicefrac32], we construct $ u_0\in L^\infty(\mathbb R)\cap\dot H^1(\mathbb R)\cap\dot H^s(\mathbb R) $ whose unique global dissipative solution satisfies u∉ C([0,T]; Ḣs (ℝ)), for every T>0. The constructions differ substantially in the subcritical and critical regimes. For 1<s<nicefrac32, we superpose rescaled, localized bubbles with increasingly negative slopes; subcritical scaling preserves Ḣs-summability, while the explicit characteristic formula produces norm inflation. At s=nicefrac32, where scaling yields no smallness, we use logarithmically distributed compactly supported multiscale profiles whose breaking times converge to zero. A one-sided localization principle and an almost-orthogonality estimate then transfer the inflation of individual profiles to the full dissipative solution.
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BibTeX
@misc{guelmame2026strong,
title = {{Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation}},
author = {Billel Guelmame and Haroune Houamed},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02523},
eprint = {2609.02523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02523},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation
AU - Billel Guelmame
AU - Haroune Houamed
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02523
UR - https://arxiv.org/abs/2609.02523
AB - We study the Hunter--Saxton equation on the real line and its global dissipative solution in the energy space $ L^\infty(\mathbb R)\cap \dot H^1(\mathbb R). $ We prove strong ill-posedness through instantaneous failure of Sobolev regularity at and below the Lipschitz threshold. More precisely, for every s∈(1,nicefrac32], we construct $ u_0\in L^\infty(\mathbb R)\cap\dot H^1(\mathbb R)\cap\dot H^s(\mathbb R) $ whose unique global dissipative solution satisfies u∉ C([0,T]; Ḣ^s (ℝ)), for every T>0. The constructions differ substantially in the subcritical and critical regimes. For 1<s<nicefrac32, we superpose rescaled, localized bubbles with increasingly negative slopes; subcritical scaling preserves Ḣ^s-summability, while the explicit characteristic formula produces norm inflation. At s=nicefrac32, where scaling yields no smallness, we use logarithmically distributed compactly supported multiscale profiles whose breaking times converge to zero. A one-sided localization principle and an almost-orthogonality estimate then transfer the inflation of individual profiles to the full dissipative solution.
ER -