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Dispersive decay for the Inter-critical nonlinear Schrödinger equation in $\mathbb{R}^3$

Boyu Jiang, Jiawei Shen, Kexue Li

Abstract

This paper investigates the Cauchy problem for the nonlinear Schrödinger equation (NLS) in the mass-supercritical and energy-subcritical regime within three spatial dimensions. For initial data in the critical homogeneous Sobolev space $\dot{H}^{s_c}(\mathbb{R}^3)$ (where $s_c = \frac{5}{6}$), we get a uniform decay estimate for the long-time dynamics of solutions, which extends the previous results.

Dispersive decay for the Inter-critical nonlinear Schrödinger equation in $\mathbb{R}^3$

Abstract

This paper investigates the Cauchy problem for the nonlinear Schrödinger equation (NLS) in the mass-supercritical and energy-subcritical regime within three spatial dimensions. For initial data in the critical homogeneous Sobolev space (where ), we get a uniform decay estimate for the long-time dynamics of solutions, which extends the previous results.
Paper Structure (9 sections, 13 theorems, 72 equations)

This paper contains 9 sections, 13 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. some setting and bascial Lemma
  3. Homogeneous Sobolev Space
  4. Lorentz Space
  5. Lorentz–Strichartz estimates
  6. Lorentz spacetime bounds
  7. Proof of Theorem \ref{['main thm']}
  8. Integrable case
  9. Non-integrable case

Key Result

Theorem 1.1

(Well-posedness) Let $d \ge 1$, and $u_0(x) \in H^{s_c}$. Furthermore, assume $0 \le s_c \le 1$. If $\||\nabla|^{s_c}u_0\|_{L^2} < \delta(d)$ is sufficiently small, then $u(t)$ does not blow up forward or backward in time (global existence holds). That is, scattering holds, and where $(q, r)$ is a Schrödinger admissible pair.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Bernstein inequality
  • Definition 2.2: Lorentz space
  • Lemma 2.3: Hunt interpolation
  • Lemma 2.4: Hölder's inequality
  • Lemma 2.5: Young--O'Neil convolutional inequality
  • Lemma 2.6: Sobolev embedding
  • Lemma 2.7: Leibniz rule
  • Definition 2.8: Schrödinger-admissible
  • ...and 10 more