Lower-dispersion global theory in two dimensions

Extend the global well-posedness result for the two-dimensional cubic BBM equation to lower values of the fractional dispersion parameter \beta within the range 1\leq\beta<2.

Background

For the two-dimensional cubic BBM equation, the paper proves global well-posedness in Hs(T2) for \beta=2 and s\geq 1/2. The author discusses attempts to adapt the energy and high–low frequency methods to smaller dispersion exponents, but these approaches do not yield a result within the admissible range. The extension to lower \beta therefore remains unresolved.

References

It is not clear to the author how to extend this result to a lower range of $\beta$.

— Well-posedness for the generalised fractional BBM equation  (2610.03288 - Barratt, 2 Oct 2026) in Remark \ref{Dim_two_restriction_remark}, subsection “Dimension 2”