发表机构
Monash University（莫纳什大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究广义分数阶BBM方程在环面上的适定性，改进低正则性结果，证明三次正则化Benjamin-Ono方程的端点全局适定性，并展示负正则性下的强不适定性。
AI 中文摘要
本文研究了在$d$维环面$\mathbb{T}^d$（$d\geq 1$）上的一族广义分数阶BBM方程。我们在$H^s(\mathbb{T}^d)$尺度上建立了低正则性的局部和全局适定性结果，这些结果改进了Bona和Chen（2003）以及Kim和Kwak（2026）的工作。我们还证明了在$H^{1/2}(\mathbb{T})$中三次正则化Benjamin-Ono方程的端点全局适定性。此外，我们在$L^{k+1}(\mathbb{T}^d)$空间中获得了广义分数阶BBM的适定性结果，其中$k$是多项式非线性的阶数。作为该结果的一个应用，我们证明对于一定范围的参数值以及对初始数据进行适当的随机化，该模型在$H^s(\mathbb{T}^d)$中（$s\geq 0$）几乎必然全局适定。最后，我们证明该族在$H^s(\mathbb{T}^d)$（$s<0$）中表现出强不适定性，具体表现为所有初始数据处正则性的无限损失。
英文摘要
In this paper, we study a generalised family of fractional BBM equations on the $d$ dimensional torus $\mathbb{T}^d$ for $d\geq 1$. We establish low regularity local and global well-posedness results on the $H^s(\mathbb{T}^d)$ scale that improve upon the works of Bona and Chen (2003) and Kim and Kwak (2026). We also prove endpoint global well-posedness for the cubic regularised Benjamin-Ono equation in $H^{1/2}(\mathbb{T})$. Furthermore, we obtain a well-posedness result for the generalised fractional BBM in the spaces $L^{k+1}(\mathbb{T}^d)$ where $k$ is order of the polynomial nonlinearity. As an application of this result, we show that for a certain range of values and for a suitable randomisation of the initial data, the model is almost surely globally well-posed in $H^s(\mathbb{T}^d)$ for $s\geq 0$. Lastly, we show that the family exhibits a strong form of ill-posedness in $H^s(\mathbb{T}^d)$ for $s<0$ in the form of an infinite loss of regularity at all initial datum.
Comments46 Pages
