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Threshold solutions of the energy-critical complex Ginzburg-Landau equation

Xing Cheng, Yunrui Zheng

Abstract

In this article, we consider energy-critical complex Ginzburg-Landau equation in three and four dimensions. We give the dynamics when the energy of the initial data is equal to the energy of the stationary solution.

Threshold solutions of the energy-critical complex Ginzburg-Landau equation

Abstract

In this article, we consider energy-critical complex Ginzburg-Landau equation in three and four dimensions. We give the dynamics when the energy of the initial data is equal to the energy of the stationary solution.
Paper Structure (5 sections, 10 theorems, 54 equations)

This paper contains 5 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. The case (A)
  4. The case (B)
  5. The case (C)

Key Result

Theorem 1.1

Let $u_0 \in \dot{H}^1(\mathbb{R}^d )$ and $E(u_0) < E(W)$, where $W$ is the stationary solution of eq:gl given by eq1.2v11. Then we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1: Sharp Sobolev inequality
  • Lemma 2.2: Unconditional uniqueness
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 7 more