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

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

2008-08-18

arxiv.org/abs/0808.2324v1

Ann. Henri Poincar\'e 10 (2009), no. 6, 1159-1205

Mathematics

Analysis of PDEs

39 pages

Scientific paper

10.1007/s00023-009-0010-2

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.

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