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Dispersive decay for the energy-critical nonlinear Schrödinger equation

Matthew Kowalski

TL;DR

This work establishes pointwise-in-time dispersive decay for the energy-critical nonlinear Schrödinger equation in dimensions $d=3,4$ for both initial-value and final-state problems, with data in scaling-critical spaces. The authors fuse Lorentz-Strichartz estimates, Lorentz-space spacetime bounds, and Besov-space techniques to control the nonlinear term at critical regularity, achieving nonlinear decay rates $|t|^{-d(\frac12-\frac1p)}$ consistent with linear decay. In $d=4$, they further obtain full $L^\infty_x$ decay for data in the Besov space $\dot{B}^1_{2,1}$ and derive corollaries for interpolation between Sobolev spaces, leveraging a Besov paraproduct framework and stability results. The final-state analysis shows that dispersive decay persists through the interaction time, with the results extending the nonlinear dispersive theory to scaling-critical settings and advancing understanding of long-time dynamics for energy-critical NLS.

Abstract

We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schrödinger equation in spatial dimensions $d = 3,4$ for both the initial-value and final-state problems.

Dispersive decay for the energy-critical nonlinear Schrödinger equation

TL;DR

This work establishes pointwise-in-time dispersive decay for the energy-critical nonlinear Schrödinger equation in dimensions for both initial-value and final-state problems, with data in scaling-critical spaces. The authors fuse Lorentz-Strichartz estimates, Lorentz-space spacetime bounds, and Besov-space techniques to control the nonlinear term at critical regularity, achieving nonlinear decay rates consistent with linear decay. In , they further obtain full decay for data in the Besov space and derive corollaries for interpolation between Sobolev spaces, leveraging a Besov paraproduct framework and stability results. The final-state analysis shows that dispersive decay persists through the interaction time, with the results extending the nonlinear dispersive theory to scaling-critical settings and advancing understanding of long-time dynamics for energy-critical NLS.

Abstract

We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schrödinger equation in spatial dimensions for both the initial-value and final-state problems.

Paper Structure

This paper contains 10 sections, 21 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Lorentz theory
  3. Lorentz--Strichartz estimates
  4. Lorentz spacetime bounds
  5. Proof of Theorem \ref{['theorem']}
  6. Integrable case
  7. Non-integrable case
  8. Besov theory
  9. Proof of Theorem \ref{['edge case theorem']}
  10. Final-state problem

Key Result

Theorem 1.1

Fix $d \geq 3$ and let $u_0 \in \dot{H}^1(\mathbb{R}^d)$. In the focusing case, assume that $u_0$ satisfies $\|u_0\|_{\dot{H}^1} < \|W\|_{\dot{H}^1}$ and $E(u_0) < E(W)$. In the $d = 3$ focusing case, further assume that $u_0$ is radial. Then there exists a unique global solution $u \in C_t\dot{H}_x Moreover, there exist scattering states $u_{\pm} \in \dot{H}^1$ such that

Theorems & Definitions (37)

  • Theorem 1.1: Well-posedness
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Lorentz space
  • Lemma 2.2: Hunt interpolation
  • Lemma 2.3: Hölder's inequality
  • Lemma 2.4: Young--O'Neil convolutional inequality
  • Lemma 2.5: Sobolev embedding
  • ...and 27 more