An improvement on the Brézis–Gallouët technique for 2D NLS and 1D half-wave equation

Abstract

We revise the classical approach by Brézis–Gallouët to prove global well-posedness for nonlinear evolution equations. In particular we prove global well-posedness for the quartic NLS on general domains Ω in ${\mathbb{R}}^2$ with initial data in $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$, and for the quartic nonlinear half-wave equation on $\mathbb{R}$ with initial data in $H^1(\mathbb{R})$.