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Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

Xueying Yu, Haitian Yue, Zehua Zhao

Abstract

In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) $\mathbb{R}^d\times \mathbb{T}^n$, $d \geq 5$, $n=1,2,3$. The tori component $\mathbb{T}^n$ can be generalized to $n$-dimensional compact manifolds $\mathcal{M}^n$. First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher dimensional analogue, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader \cite{Pau2,Pau1,Pau3} and the 4NLS analogue of Tzvetkov-Visciglia \cite{TV2}.

Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

Abstract

In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) , , . The tori component can be generalized to -dimensional compact manifolds . First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher dimensional analogue, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader \cite{Pau2,Pau1,Pau3} and the 4NLS analogue of Tzvetkov-Visciglia \cite{TV2}.
Paper Structure (19 sections, 12 theorems, 135 equations)

This paper contains 19 sections, 12 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Background
  4. Organization of the rest of this paper
  5. Notations
  6. Useful estimates and admissible exponents
  7. Well-posedness theory
  8. Morawetz estimates
  9. Decay property and Scattering
  10. Step 1: Proof of the decay property
  11. Step 2: Proof of the spacetime bound
  12. Step 3: Proof of the scattering asymptotics
  13. Discussions regarding the higher dimensional analogue and the focusing scenario
  14. The higher dimensional analogue
  15. The focusing scenario
  16. ...and 4 more sections

Key Result

Theorem 1.4

The initial value problem maineq has a unique local solution $u(t,x,\alpha)\in \mathcal{C}((-T,T);H_{x,\alpha}^{2} (\mathbb{R}^d\times \mathbb{T}))$ where $T=T(\|u_0\|_{H_{x,\alpha}^{2}}(\mathbb{R}^d\times \mathbb{T}))>0$; moreover, the solution $u(t, x, \alpha)$ can be extended globally in time by

Theorems & Definitions (28)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 18 more