Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SMALL DATA SCATTERING OF HARTREE TYPE FRACTIONAL SCHRÖDINGER EQUATIONS IN DIMENSION 2 AND 3

  • Cho, Yonggeun (Department of Mathematics and Institute of Pure and Applied Mathematics Chonbuk National University) ;
  • Ozawa, Tohru (Department of Applied Physics Waseda University)
  • Received : 2017.04.02
  • Accepted : 2017.07.24
  • Published : 2018.03.01
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Abstract

In this paper we study the small-data scattering of the d dimensional fractional $Schr{\ddot{o}}dinger$ equations with d = 2, 3, $L{\acute{e}}vy$ index 1 < ${\alpha}$ < 2 and Hartree type nonlinearity $F(u)={\mu}({\mid}x{\mid}^{-{\gamma}}{\ast}{\mid}u{\mid}^2)u$ with max(${\alpha}$, ${\frac{2d}{2d-1}}$) < ${\gamma}{\leq}2$, ${\gamma}$ < d. This equation is scaling-critical in ${\dot{H}}^{s_c}$, $s_c={\frac{{\gamma}-{\alpha}}{2}}$. We show that the solution scatters in $H^{s,1}$ for any s > $s_c$, where $H^{s,1}$ is a space of Sobolev type taking in angular regularity with norm defined by ${\parallel}{\varphi}{\parallel}_{H^{s,1}}={\parallel}{\varphi}{\parallel}_{H^s}+{\parallel}{\nabla}_{{\mathbb{S}}{\varphi}}{\parallel}_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere ${\mathbb{S}}^{d-1}$ and utilize $U^p-V^p$ space argument.

Keywords

Acknowledgement

Supported by : Chonbuk National University

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