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Soliton resolution for the energy-critical nonlinear Ginzburg-Landau equation in the radial case

Yuchen Yin

Abstract

We study the the energy critical non-linear Ginzburg-Landau equation $\partial_{t} u =zΔu+z|u|^{\frac{4}{D-2}} u$ with $\Re z >0$ in dimension $D\geq 3$. We prove that every radial solution with finite energy norm resolves into a finite superposition of asymptotically decoupled copies of the ground state and free radiation continuously in time.

Soliton resolution for the energy-critical nonlinear Ginzburg-Landau equation in the radial case

Abstract

We study the the energy critical non-linear Ginzburg-Landau equation with in dimension . We prove that every radial solution with finite energy norm resolves into a finite superposition of asymptotically decoupled copies of the ground state and free radiation continuously in time.
Paper Structure (20 sections, 22 theorems, 135 equations)

This paper contains 20 sections, 22 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Setting of the Problem
  3. Main Results
  4. Strategy of the Proof
  5. Notation
  6. Acknowledgment
  7. Preliminaries
  8. Local Cauchy Theory
  9. Linearization Around a Ground State
  10. Multi-Bubble Configuration
  11. Localized Energy Inequalities and Energy Trapping
  12. The Sequential Decomposition
  13. Identification of the Body Map
  14. Localized Sequential Bubbling
  15. The Sequential Decomposition for Finite Time Blow-Up Solutions
  16. ...and 5 more sections

Key Result

Theorem 1

Let $D \geq 3$ , and let ${u}(t)$ be the solution to eqn:GL with initial data $u(0)=u_0\in \mathcal{E}$, defined on its maximal interval of existence $[0, T_{+})$. Suppose that then (Global solution) If $T_{+}=\infty$, there exist a time $T_0>0$, an integer $N \geq 0$, continuous functions $\lambda_1(t), \ldots, \lambda_N(t) \in C^0\left([T_0, \infty),(0,\infty)\right)$, phases $\theta_1(t), \ldo

Theorems & Definitions (52)

  • Theorem 1: Soliton resolution
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4: Multi-bubble configuration
  • Definition 1.5
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • ...and 42 more