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Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation

Xing Cheng, Chang-Yu Guo, Yunrui Zheng

TL;DR

This work addresses the long-standing question of global weak solutions for the 3D focusing energy-critical NLS in the non-radial setting by leveraging energy-critical Ginzburg-Landau equations as dissipative approximations. It develops a concentration-compactness/rigidity framework to exclude a minimal-energy blowup (critical) element and derives global weak solutions via a limiting argument, complemented by a weak-strong uniqueness result under a subthreshold data constraint relative to the ground state $W$. The authors also prove asymptotic decay to zero for subthreshold global solutions and provide an inviscid-limit construction of weak solutions together with a stability result against strong solutions. Overall, the paper bridges dissipative GL dynamics and dispersive NLS, advancing the understanding of focusing energy-critical flows in non-radial 3D and offering a rigorous mechanism to obtain well-posedness and stability results in this challenging regime.

Abstract

In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schrödinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Schördinger equation. In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1( \mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schrödinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is bounded by a constant depending on the stationary solution $W$.

Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation

TL;DR

This work addresses the long-standing question of global weak solutions for the 3D focusing energy-critical NLS in the non-radial setting by leveraging energy-critical Ginzburg-Landau equations as dissipative approximations. It develops a concentration-compactness/rigidity framework to exclude a minimal-energy blowup (critical) element and derives global weak solutions via a limiting argument, complemented by a weak-strong uniqueness result under a subthreshold data constraint relative to the ground state . The authors also prove asymptotic decay to zero for subthreshold global solutions and provide an inviscid-limit construction of weak solutions together with a stability result against strong solutions. Overall, the paper bridges dissipative GL dynamics and dispersive NLS, advancing the understanding of focusing energy-critical flows in non-radial 3D and offering a rigorous mechanism to obtain well-posedness and stability results in this challenging regime.

Abstract

In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schrödinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Schördinger equation. In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in for when the energy is less than the energy of the stationary solution . We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schrödinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the -norm of the initial data is bounded by a constant depending on the stationary solution .
Paper Structure (14 sections, 24 theorems, 245 equations)

This paper contains 14 sections, 24 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Local well-posedness theory
  4. Some variational facts of Ginzburg-Landau equation
  5. Proof of Blow-up
  6. Existence of a critical element
  7. Rigidity
  8. Preliminary tools
  9. Proof of Theorem \ref{['thm:rigidity']}
  10. Asymptotic decay of global solutions
  11. Proof of Theorem \ref{['th1.2v21']}
  12. Existence of global weak solutions
  13. Weak-strong uniqueness
  14. Remarks on the Galerkin method

Key Result

Theorem 1.2

Let $v_0 \in \dot{H}^1$, when $\mu = - 1$, the corresponding solution $v$ to eq1.1v18 is global for $d \ge 3$. When $\mu = 1$ and $E(v_0) < E(W)$, where the following conclusions hold: Furthermore, the global solution scatters in $\dot{H}^1$.

Theorems & Definitions (45)

  • Theorem 1.2: GWP & scattering of the energy-critical nonlinear Schrödinger equation
  • Definition 1.1
  • Theorem 1.2: Global weak solution of the 3-dimensional focusing energy-critical NLS
  • Remark 1.3
  • Theorem 1.4
  • Definition 2.1: Strong solution
  • Lemma 2.2
  • Theorem 2.3: Local well-posedness
  • Proposition 2.4: Stability
  • Proposition 2.5
  • ...and 35 more