Global well-posedness for the modified dispersion-generalized Benjamin-Ono equations

Abstract

In this short note, we prove the global well-posedness of the modified dispersion-generalized Benjamin-Ono equation with dispersion $1<\alpha <2$, $${\partial }_{t}u+D^{\alpha }{\partial }_{x}u+u^2{\partial }_{x}u=0,$$ over the Sobolev space $H^s(\mathbb{R})$, for any regularity $s$ greater or equal than the sharp analytic well-posedness regularity. As a byproduct, we also derive nonlinear smoothing estimates and give a simple proof of local well-posedness.

arXiv comment12 pages