Blow up dynamics for the 3D energy-critical Nonlinear Schr\"odinger equation

TL;DR

This paper constructs a continuum of finite-time blow-up solutions for the 3D energy-critical focusing nonlinear Schrödinger equation, describing their precise asymptotic behavior near the blow-up time.

Contribution

It introduces a new family of type II blow-up solutions with detailed asymptotics for the 3D energy-critical NLS, expanding understanding of blow-up dynamics.

Findings

Constructed a two-parameter family of blow-up solutions.

Described the precise asymptotic form near blow-up time.

Showed solutions collapse to a single energy bubble.

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) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}}$ is the ground state solution, $\lambda(t) = (T-t)^{- \frac12 - \nu} $ for suitable $ \nu > 0 $, $ \alpha(t) = \alpha_0 \log(T - t)$ and $ T= T(\nu, \alpha_0) > 0 $. Further $ \|\eta(t) - \eta_T\|_{\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$.