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Global well-posedness and scattering for the defocusing $\dot{H}^{\frac{1}{2}}$-critical nonlinear Schrödinger equation in $\mathbb{R}^2$

Xueying Yu

TL;DR

This work proves global well-posedness and scattering for the defocusing quintic NLS in two dimensions at the $\dot{H}^{1/2}$-critical level. The authors adapt a concentration-compactness framework, constructing a minimal counterexample that is almost periodic modulo symmetries and then ruling it out via a novel long-time Strichartz theory built around the $\tilde{X}_{k_0}$-norm and atomic spaces. A frequency-localized Morawetz estimate complements this analysis, excluding quasi-soliton scenarios and establishing global scattering with an explicit a priori bound on the spacetime integral $\iint |u|^8 dx dt$. The combination of profile decomposition, upside-down long-time control, and rigidity arguments extends the critical $\dot{H}^{1/2}$ theory to the two-dimensional, non-endpoint setting, providing sharp insight into the critical dynamics of 2D NLS.

Abstract

In this paper we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schrödinger equation in $\mathbb{R}^2$ with general data in the critical space $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$. We show that if a solution remains bounded in $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$ in its maximal interval of existence, then the interval is infinite and the solution scatters.

Global well-posedness and scattering for the defocusing $\dot{H}^{\frac{1}{2}}$-critical nonlinear Schrödinger equation in $\mathbb{R}^2$

TL;DR

This work proves global well-posedness and scattering for the defocusing quintic NLS in two dimensions at the -critical level. The authors adapt a concentration-compactness framework, constructing a minimal counterexample that is almost periodic modulo symmetries and then ruling it out via a novel long-time Strichartz theory built around the -norm and atomic spaces. A frequency-localized Morawetz estimate complements this analysis, excluding quasi-soliton scenarios and establishing global scattering with an explicit a priori bound on the spacetime integral . The combination of profile decomposition, upside-down long-time control, and rigidity arguments extends the critical theory to the two-dimensional, non-endpoint setting, providing sharp insight into the critical dynamics of 2D NLS.

Abstract

In this paper we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schrödinger equation in with general data in the critical space . We show that if a solution remains bounded in in its maximal interval of existence, then the interval is infinite and the solution scatters.

Paper Structure

This paper contains 26 sections, 42 theorems, 218 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. A more formal outline of the proof
  3. Preliminaries
  4. Littlewood-Paley theory
  5. Estimates from harmonic analysis
  6. Strichartz estimates
  7. Profile decomposition
  8. Local theory and stability
  9. Impossibility of finite-time blow-up solutions
  10. Atomic spaces and $X$-norm
  11. Basic definitions and properties of the atomic space
  12. $\tilde{X}_{k_0}$-norm
  13. Choice of $\varepsilon_1$, $\varepsilon_2$ and $\varepsilon_3$
  14. Definitions of small intervals: $J_l$ and $J^{\alpha}$
  15. Construction of $G_k^j$ intervals and restrictions of $\varepsilon_1$, $\varepsilon_2$ and $\varepsilon_3$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $u : I \times \mathbb{R}^2 \to \mathbb{C}$ be a maximal-lifespan solution to NLS such that $u \in L_t^{\infty} \dot{H}_x^{\frac{1}{2}} (I \times \mathbb{R}^2 )$. Then $u$ is global and scatters (see Definition defn Scattering), with for some function $C : [0, \infty) \to [0,\infty)$.

Figures (1)

  • Figure 4.1: $\tilde{X}_{k_0}$ norm

Theorems & Definitions (85)

  • Theorem 1.1: Main theorem
  • Definition 1.2: Solution
  • Definition 1.3: Scattering
  • Definition 1.4: Scattering size and blow up
  • Theorem 1.5: Local well-posedness
  • Theorem 1.6: Existence of minimal counterexamples
  • Definition 1.7: Almost periodicity
  • Lemma 1.8: Local constancy of $N(t)$ and $x(t)$
  • Remark 1.9
  • Corollary 1.10: $N(t)$ at blow-up
  • ...and 75 more