Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

Youngwoo Koh; Ihyeok Seo

doi:10.3934/dcds.2017210

Article Contents

2017, Volume 37, Issue 9: 4877-4906. Doi: 10.3934/dcds.2017210

This issue Previous Article General decay of solutions of a Bresse system with viscoelastic boundary conditions Next Article Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two

Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

Youngwoo Koh^1, and
Ihyeok Seo^2, ,

1.
Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea
2.
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

* Corresponding author: Ihyeok Seo
* Corresponding author: Ihyeok Seo

Received: April 30, 2016

Revised: March 31, 2017

Published: October 2017

Y. Koh was supported by NRF Grant 2016R1D1A1B03932049 (Republic of Korea). I. Seo was supported by the TJ Park Science Fellowship of POSCO TJ Park Foundation and by the NRF grant funded by the Korea government(MSIP) (No. 2017R1C1B5017496)

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Abstract

We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-\Delta)^{a/2}u=F(x, t)$, $u(x, 0)=f(x)$, of general orders $a>1$ with radial data $f, F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in [1] concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted $L^2$ estimates to the well-posedness theory for the Schrödinger equations with time-dependent potentials in the class.

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Mathematics Subject Classification: Primary: 35B45, 35A01; Secondary: 35Q40, 42B35.

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Figure 1. The region of $(s,1/p)$ for (7) particularly when $a=2$

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