Sharpness of the local theory in higher dimensions

Determine whether the local well-posedness threshold for the generalised fractional BBM equation on the torus is sharp in dimensions d\geq 2.

Background

The paper proves local well-posedness for the generalised fractional BBM equation on the d-dimensional torus under an explicit Sobolev regularity condition. In dimension one, the author combines existing ill-posedness results with arguments adapted from prior work to establish sharpness throughout the full range 1<\beta\leq 2. The corresponding sharpness question in dimensions d\geq 2 is not resolved.

References

As far as the author is aware, the sharpness of the local theory for $d\geq 2$ is still an open problem.

— Well-posedness for the generalised fractional BBM equation  (2610.03288 - Barratt, 2 Oct 2026) in Section 1, subsection “Local theory”