Global Well-Posedness for the Benjamin-Ono Equation with Small Periodic Initial Data in Analytic Spaces

Abstract

We establish global well-posedness of the Benjamin--Ono equation for sufficiently small, zero-mean periodic initial data in the analytic Sobolev space $H_0^{\rho ,1}$. The proof combines regularized commuting flows with an exponential spectral energy. A local uniform convexity property of this energy upgrades weak endpoint compactness to strong continuity, yielding a global flow in $C(\mathbb{R};H_0^{\rho ,1})$ and continuous dependence in the same analytic topology, with no dynamic loss of the prescribed analytic width.

23 pages