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Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space

Junyong Zhang, Jiqiang Zheng

Abstract

In this paper, we study Strichartz estimates for the Schrödinger equation on a metric cone $X$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. For the metric $g$ on $X$ given by $g=dr^2+r^2h$, let $Δ_g$ be the positive Friedrichs extension Laplacian on $X$ and $V=V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ is a real function such that the operator $P:=Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. We establish the full range of global-in-time Strichartz estimates without loss for the Schrödinger equation associated with the operator $\LL_V=Δ_g+V_0 r^{-2}$ including the endpoint estimate both in homogeneous and inhomogeneous cases. A new finding reveals that the range of admissible pairs at $\dot H^s$-level is influenced by the smallest eigenvalue of the operator $P$. This additionally proves the conjecture in Wang [Ann. Inst. Fourier 2006] and generalizes the results of Ford [Comm. Math. Phys. 2010] and Baskin-Marzuola-Wunsch [Contemp. Math. 2014]. As an application, we show the well-posedness theory and scattering theory for the Schrödinger equation with a cubic nonlinearity on this setting which verifies a conjecture in Baskin-Marzuola-Wunsch [Contemp. Math. 2014].

Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space

Abstract

In this paper, we study Strichartz estimates for the Schrödinger equation on a metric cone , where and the cross section is a -dimensional closed Riemannian manifold . For the metric on given by , let be the positive Friedrichs extension Laplacian on and where is a real function such that the operator is a strictly positive operator on . We establish the full range of global-in-time Strichartz estimates without loss for the Schrödinger equation associated with the operator including the endpoint estimate both in homogeneous and inhomogeneous cases. A new finding reveals that the range of admissible pairs at -level is influenced by the smallest eigenvalue of the operator . This additionally proves the conjecture in Wang [Ann. Inst. Fourier 2006] and generalizes the results of Ford [Comm. Math. Phys. 2010] and Baskin-Marzuola-Wunsch [Contemp. Math. 2014]. As an application, we show the well-posedness theory and scattering theory for the Schrödinger equation with a cubic nonlinearity on this setting which verifies a conjecture in Baskin-Marzuola-Wunsch [Contemp. Math. 2014].

Paper Structure

This paper contains 22 sections, 28 theorems, 313 equations.

Table of Contents

  1. Introduction and Statement of Main Result
  2. The setting model
  3. The Strichartz estimates
  4. The main results on Strichartz estimates
  5. The Strichartz estimates at $\dot H^s$-level
  6. Application to NLS
  7. Preliminary: analysis results associated with ${\mathcal{L}}_V$
  8. Hankel transform and Bessel function
  9. $L^p$-product chain rule
  10. Hardy inequality
  11. Spectral measure and Strichartz estimates associated with $\mathcal{L}_0$
  12. Spectral measure
  13. Strichartz estimates for the case $V=0$
  14. Local-smoothing and Strichartz estimates
  15. The endpoint inhomogeneous Strichartz estimate
  16. ...and 7 more sections

Key Result

Theorem 1.1

Suppose that $(X, g)$ is a metric cone of dimension $n\geq3$. Let ${\mathcal{L}}_V=\Delta_g+V$ where $r^2V:=V_0\in{\mathcal{C}}^\infty(Y)$ such that $\Delta_h+V_0(y)+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. Then the homogenous Strichartz estimates holds for the admissible pair $(q,{\mathrm{r}})\in [2,\infty]^2$ satisfies adm-p. Moreover, the inhomogeneous inequality holds for admi

Theorems & Definitions (65)

  • Theorem 1.1: Global-in-time Strichartz estimates
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.3
  • Remark 1.6
  • Lemma 2.1: Asymptotics of the Bessel function
  • ...and 55 more