PreprintCC BY 4.0arXiv
Almost sure global well-posedness for the Benjamin-Bona-Mahony equation
Abstract
We prove that the Benjamin-Bona-Mahony equation is almost surely globally well-posed with respect to random Gaussian initial data of negative Sobolev regularity in Hs(𝕋) for any s>−1/4. This result is sharp in view of the mild probabilistic ill-posedness due to Oh-Tzvetkov (2026). This also improves on a previous result of the author which established this only in a logarithmically negative regularity. To break through this logarithmic regularity barrier, we combine the I-method with the low-high argument from Bona-Tzvetkov (2007), which motivates a refined first-order expansion for solutions.
Related
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BibTeX
@misc{forlano2026almost,
title = {{Almost sure global well-posedness for the Benjamin-Bona-Mahony equation}},
author = {Justin Forlano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36403},
eprint = {2609.36403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36403},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Almost sure global well-posedness for the Benjamin-Bona-Mahony equation
AU - Justin Forlano
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36403
UR - https://arxiv.org/abs/2609.36403
AB - We prove that the Benjamin-Bona-Mahony equation is almost surely globally well-posed with respect to random Gaussian initial data of negative Sobolev regularity in H^s(𝕋) for any s>−1/4. This result is sharp in view of the mild probabilistic ill-posedness due to Oh-Tzvetkov (2026). This also improves on a previous result of the author which established this only in a logarithmically negative regularity. To break through this logarithmic regularity barrier, we combine the I-method with the low-high argument from Bona-Tzvetkov (2007), which motivates a refined first-order expansion for solutions.
ER -