An Improvement on the Br\'ezis-Gallou\"et technique for 2D NLS and 1D half-wave equation

We revise the classical approach by Br\'ezis-Gallou\"et to prove global well posedness for nonlinear evolution equations. In particular we prove global well--posedness for the quartic NLS posed on general domains $\Omega$ in $\R^2$ with initial data in $H^2(\Omega)\cap H^1_0(\Omega)$, and for the quartic nonlinear half-wave equation on $\R$ with initial data in $H^1(\R)$.