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 on general domains Ω in R2R2 with initial data in H2(Ω)∩H01(Ω), and for the quartic nonlinear half-wave equation on RR with initial data in H1(R)H1(R).