Global Well-Posedness of the defocussing Davey-Stewartson II Equation: The Work of Nachman-Regev-Tataru

The defocussing Davey-Stewartson II equation is a dispersive nonlinear equation which describes weakly nonlinear surface waves on a two-dimensional surface. In a remarkable paper (see https://arxiv.org/abs/1708.04759), Nachman, Regev, and Tataru use ideas of inverse scattering and harmonic analysis to prove global well-posedness and scattering in $L^2$. We'll describe key elements of their result including (i) new estimates on fractional integrals and pseudo differential operators, (ii) resolvent bounds via concentration compactness, and (iii) Lipschitz estimates on the scattering transform. A key role is played in the estimates by the Hardy-Littlewood maximal function.