Blow up dynamics for the 3D energy-critical Nonlinear Schrödinger equation

Abstract

We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $d=3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $u(t,x)=e^{i\alpha (t)}\lambda (t{)}^{\frac{1}{2}}W(\lambda (t)x)+\eta (t,x)$, $t\in [0,T)$, $x\in {\mathbb{R}}^3$, where $W(x)=(1+\frac{|x{|}^2}{3}{)}^{-\frac{1}{2}}$ is the ground state solution, $\lambda (t)=(T-t{)}^{-\frac{1}{2}-\nu }$ for suitable $\nu >0$, $\alpha (t)={\alpha }_0\log (T-t)$ and $T=T(\nu ,{\alpha }_0)>0$. Further $\Vert \eta (t)-{\eta }_T{\Vert }_{{\dot{H}}^1\cap {\dot{H}}^2}=o(1)$ as $t\to T^{-}$ for some ${\eta }_T\in {\dot{H}}^1\cap {\dot{H}}^2$.

arXiv comment123 pages, minor revision of Proposition 5.4 (numerical verification), typos corrected, footnotes partially removed