On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS

We consider L2-critical focusing nonlinear Schrödinger equations with Hartree type nonlinearity
i \partial_{t} u = - \Delta u - \left(\Phi \ast |u|^2 \right) u \quad {\rm in}\, \mathbb {R}^4,
where Φ(x) is a perturbation of the convolution kernel |x|−2. Despite the lack of pseudo-conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions u(t, x) that exhibit the pseudo-conformal blowup rate
\| \nabla u(t) \|_{L^2_x}\sim \frac{1}{|t|} \quad {\rm as}\, t \nearrow 0.
Furthermore, we prove the finite-codimensional stability of this conformal blow up, by extending the nonlinear wave operator construction by Bourgain and Wang (see Bourgain and Wang in Ann. Scuola Norm Sup Pisa Cl Sci (4) 25(1–2), 197–215, 1997/1998) to L2-critical Hartree NLS.
Communicated by Rafael D. Benguria.