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

Tobias Schmid

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$.

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 . The solutions collapse to a single energy bubble in finite time, precisely they have the form , , , where is the ground state solution, for suitable , and . Further as for some .
Paper Structure (21 sections, 77 theorems, 758 equations, 1 figure)

This paper contains 21 sections, 77 theorems, 758 equations, 1 figure.

Table of Contents

  1. Introduction & Notation
  2. Brief overview of the Cauchy problem
  3. Notation and Outline
  4. Construction of approximate solutions near the ground state
  5. Interior region $r \lesssim (T -t)^{\frac{1}{2} + \epsilon_1 }$
  6. The first two corrections
  7. The general induction step
  8. Self-similar region $(T -t)^{\frac{1}{2} + \epsilon_1 } \lesssim r \lesssim (T -t)^{\frac{1}{2} - \epsilon_2 }$
  9. The asymptotic behavior at $y \gg1$
  10. Remote region $(T -t)^{\frac{1}{2} - \epsilon_2 } \lesssim r$
  11. The final approximation
  12. Completion to exact solutions: Final iteration
  13. The final system
  14. Scattering theory and the distorted Fourier side for Schrödinger systems
  15. Scattering Theory
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\alpha_0 \in \mathop{\mathrm{\mathbb{R}}}\nolimits$, $d = 3$ and $~ \nu > 1$. Then there exists $M = M(\alpha_0, \nu) > 0$ small such that the following holds. Let $\delta \in (0,M)$. Then for some small $T_0 >0$ and any $T \in (0, T_0)$, there is a solution $u(t) = u_{T, \alpha_0, \nu}(t)$ of where $\lambda(t) = (T-t)^{- \frac{1}{2}- \nu},~\alpha(t) = \alpha_0 \log(T-t)$ and $\eta \in C^0([

Figures (1)

  • Figure 1: The contour $\Gamma_{R, \delta}^+$

Theorems & Definitions (164)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Proposition 2.1: Approximation
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 154 more