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

Abstract

We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \pr_t u = -\DD u - \big (Φ\ast |u|^2 \big) u \quad {in $\RR^4$},$$ 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 $$\Vert \nabla u(t){\Vert }_{L_x^2}\sim \frac{1}{|t|}ast\nearrow 0.$$ 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.

39 pages

References in corpus (3)

• Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations
• Blow-Up for Nonlinear Wave Equations describing Boson Stars
• Effective Dynamics for Boson Stars

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