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

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 posed on general domains $Ω$ in $\R^2$ with initial data in $H^2(Ω)\cap H^1_0(Ω)$, and for the quartic nonlinear half-wave equation on $\R$ with initial data in $H^1(\R)$.

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