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 α(t)}λ(t)^{\frac{1}{2}}W(λ(t) x) + η(t, x )$, $ t \in[0, T)$, $ x \in \mathbb{R}^3$, where $ W( x) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}}$ is the ground state solution, $λ(t) = (T-t)^{- \frac12 - ν} $ for suitable $ ν> 0 $, $ α(t) = α_0 \log(T - t)$ and $ T= T(ν, α_0) > 0 $. Further $ \|η(t) - η_T\|_{\dot{H}^1 \cap \dot{H}^2} = o(1)$ as $ t \to T^-$ for some $ η_T \in \dot{H}^{1} \cap~ \dot{H}^2$.