  In this paper, we prove global well-posedness and scattering of the Cauchy
problem for the elliptic-elliptic Davey-Stewartson system (eeDS) for initial
data $u_{0}\in L^{2}(\mathbb{R}^{2})$ in the defocusing case and for $u_{0}\in
L^{2}(\mathbb{R}^{2})$ with mass below that of the ground state in the focusing
case. This result resolves the large data problem at the scaling-critical
regularity left open by Ghidaglia and Saut in their work initiating the
mathematical study of the Cauchy problem for the system. Our proof uses the
concentration compactness/rigidity road map of Kenig and Merle together with
the long-time Strichartz estimate approach of Dodson. Due to the failure of the
endpoint $L_{t}^{2}L_{x}^{\infty}$ Strichartz estimate, we rely heavily on
bilinear Strichartz estimates. We overcome the obstruction to applying such
estimates caused by the lack of permutation invariance of the eeDS nonlinearity
under frequency decomposition by introducing a new frequency cube decomposition
of the nonlinearity and proving bilinear estimates suited to this
decomposition. In both the defocusing and focusing cases, we overcome the lack
of an a priori interaction Morawetz estimate by exploiting the spatial and
frequency localization of the minimal counterexamples which we reduce to
considering.
