Well-posedness for the generalised fractional BBM equation

Abstract

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.