On Stability of Pseudo-Conformal Blowup for L^2-critical Hartree NLS

TL;DR

This paper demonstrates the existence and stability of pseudo-conformal blowup solutions for an $L^2$-critical Hartree NLS in four dimensions, despite the lack of conformal invariance.

Contribution

It extends the nonlinear wave operator method to establish stable blowup solutions for Hartree NLS with a perturbed kernel, a novel result in this context.

Findings

Existence of finite-time blowup solutions with pseudo-conformal rate.

Finite-codimensional stability of these blowup solutions.

Extension of Bourgain and Wang's method to Hartree NLS.

Abstract

We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \pr_t u = -\DD u - \big (\Phi \ast |u|^2 \big) u \quad {in $\RR^4$},$$ where $\Phi(x)$ 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 $$ \| \nabla u(t) \|_{L^2_x} \sim \frac{1}{|t|} \quad {as} \quad t \nearrow 0 . $$ Furthermore, we prove the finite-codimensional stability of this conformal blow up, by extending the nonlinear wave operator construction by Bourgain and Wang (see \cite{Bourgain+Wang1997}) to $L^2$-critical Hartree NLS.