Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space
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2017
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| _version_ | 1866929518616248320 |
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| author | Zhang, Junyong Zheng, Jiqiang |
| author_facet | Zhang, Junyong Zheng, Jiqiang |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1702_05813 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space Zhang, Junyong Zheng, Jiqiang Analysis of PDEs Spectral Theory 58J47, 42B37, 35Q40, 47J35 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]. |
| title | Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space |
| topic | Analysis of PDEs Spectral Theory 58J47, 42B37, 35Q40, 47J35 |
| url | https://arxiv.org/abs/1702.05813 |