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We revise the classical approach by Br\'ezis-Gallou\"et to prove global wellposedness for nonlinear evolution equations. In particular we prove globalwell--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 quarticnonlinear half-wave equation on $\R$ with initial data in $H^1(\R)$.

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