Benjamin-Bona-Mahony方程的几乎必然全局适定性
Almost sure global well-posedness for the Benjamin-Bona-Mahony equation
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中文总结 AI 辅助
本文通过结合I-方法与低-高论证，证明了Benjamin-Bona-Mahony方程在负Sobolev正则性下对随机高斯初值几乎必然全局适定，突破了先前对数正则性限制，结果达到最优。
中文摘要 AI 辅助
我们证明，对于任意 $s>-\frac 14$，Benjamin-Bona-Mahony方程在 $H^{s}(\mathbb{T})$ 中关于具有负Sobolev正则性的随机高斯初值几乎必然全局适定。鉴于Oh-Tzvetkov（2026）提出的温和概率性不适定性，这一结果是精确的。这也改进了作者先前仅在对数负正则性下建立的结果。为了突破这一对数正则性障碍，我们将 $I$-方法 与 Bona-Tzvetkov（2007）的低-高论证相结合，这激发了解的一阶展开的精细化。
英文摘要
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.
发表机构
- School of Mathematics, Monash University（蒙纳士大学数学学院）
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