A Physical-Space Approach to Modified Scattering for the 3D Hartree Equation: Full-Space and Robin Half-Space

We prove global well-posedness and modified scattering for the three-dimensional Hartree equation for small data in $H^{0,k}$, with $k>\frac{1}{2}$ on ${\mathbb{R}}^3$ and $k=1$ on the half-space ${\mathbb{R}}_{+}^3$ with a real Robin boundary condition. For these data, the nonlinear and asymptotic analysis is carried out entirely in physical space, combining slow-growth estimates for Galilean vector fields with a scale-invariant interaction estimate. For the half-space problem, we construct an extension intertwining the Robin and free flows and a boundary-adapted field that overcome the loss of normal Galilean symmetry.