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 𝕋^d for d≥ 1. We establish low regularity local and global well-posedness results on the H^s(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(𝕋). Furthermore, we obtain a well-posedness result for the generalised fractional BBM in the spaces L^k+1(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(𝕋^d) for s≥ 0. Lastly, we show that the family exhibits a strong form of ill-posedness in H^s(𝕋^d) for s<0 in the form of an infinite loss of regularity at all initial datum.

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BibTeX RIS

BibTeX

@misc{barratt2026wellposedness,
  title = {{Well-posedness for the generalised fractional BBM equation}},
  author = {Jackson Barratt},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03288},
  eprint = {2610.03288},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03288},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Well-posedness for the generalised fractional BBM equation
AU  - Jackson Barratt
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03288
UR  - https://arxiv.org/abs/2610.03288
AB  - In this paper, we study a generalised family of fractional BBM equations on the d dimensional torus 𝕋^d for d≥ 1. We establish low regularity local and global well-posedness results on the H^s(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(𝕋). Furthermore, we obtain a well-posedness result for the generalised fractional BBM in the spaces L^k+1(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(𝕋^d) for s≥ 0. Lastly, we show that the family exhibits a strong form of ill-posedness in H^s(𝕋^d) for s<0 in the form of an infinite loss of regularity at all initial datum.
ER  -

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