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arXiv · 2609.36403

Almost sure global well-posedness for the Benjamin-Bona-Mahony equation

Justin Forlano
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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 $H^{s}(\mathbb{T})$ for any $s>-\frac 14$. 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.

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Justin Forlano. 2026-09-28. Almost sure global well-posedness for the Benjamin-Bona-Mahony equation. https://arxiv.org/abs/2609.36403

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