Small data scattering of hartree type fractional schrödinger equations in a scaling critical space

Yonggeun Cho; Tohru Ozawa; Changhun Yang

doi:10.1619/fesi.64.1

Small data scattering of hartree type fractional schrödinger equations in a scaling critical space

Yonggeun Cho, Tohru Ozawa, Changhun Yang

School of Advanced Science and Engineering

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index a between 1 and 2, and Hartree type nonlinearity F(u) = µ(|x|^-y*|u|²)u with 2d/(2d - 1) < y < 2, y ≥ α > 1. This equation is scaling-critical in H^sc with s_c (y-α)/2. We show that the solution scatters in H^sc,1 where H^sc, 1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by. For this purpose we use the recently developed Strichartz estimate which is L²θ -averaged on the unit sphere S^d-1.

Original language	English
Pages (from-to)	1-15
Number of pages	15
Journal	Funkcialaj Ekvacioj
Volume	64
Issue number	1
DOIs	https://doi.org/10.1619/fesi.64.1
Publication status	Published - 2021

Keywords

Angularly averaged strichartz estimate
Hartree type fractional schrödinger equation
Scaling critical space
Small data scattering

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Geometry and Topology

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10.1619/fesi.64.1

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title = "Small data scattering of hartree type fractional schr{\"o}dinger equations in a scaling critical space",

abstract = "In this paper we study the small-data scattering of Hartree type fractional Schr{\"o}dinger equations in space dimension 2, 3. It has L{\'e}vy index a between 1 and 2, and Hartree type nonlinearity F(u) = µ(|x|-y*|u|2)u with 2d/(2d - 1) < y < 2, y ≥ α > 1. This equation is scaling-critical in Hsc with sc (y-α)/2. We show that the solution scatters in Hsc,1 where Hsc, 1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by. For this purpose we use the recently developed Strichartz estimate which is L2θ -averaged on the unit sphere Sd-1.",

keywords = "Angularly averaged strichartz estimate, Hartree type fractional schr{\"o}dinger equation, Scaling critical space, Small data scattering",

author = "Yonggeun Cho and Tohru Ozawa and Changhun Yang",

note = "Funding Information: Acknowledgements. We are grateful to the referee for an important suggestion. Y. Cho was supported by NRF-2018R1D1A3B07047782 (Republic of Korea). T. Ozawa was supported by Grant-in-Aid for Scientific Research (A) 26247014 (Japan). C. Yang was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C1005700). Publisher Copyright: {\textcopyright} 2021, KOBE UNIV. All rights reserved.",

year = "2021",

doi = "10.1619/fesi.64.1",

language = "English",

volume = "64",

pages = "1--15",

journal = "Funkcialaj Ekvacioj",

issn = "0532-8721",

publisher = "Mathematical Society of Japan - Kobe University",

number = "1",

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T1 - Small data scattering of hartree type fractional schrödinger equations in a scaling critical space

AU - Cho, Yonggeun

AU - Ozawa, Tohru

AU - Yang, Changhun

N1 - Funding Information: Acknowledgements. We are grateful to the referee for an important suggestion. Y. Cho was supported by NRF-2018R1D1A3B07047782 (Republic of Korea). T. Ozawa was supported by Grant-in-Aid for Scientific Research (A) 26247014 (Japan). C. Yang was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C1005700). Publisher Copyright: © 2021, KOBE UNIV. All rights reserved.

PY - 2021

Y1 - 2021

N2 - In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index a between 1 and 2, and Hartree type nonlinearity F(u) = µ(|x|-y*|u|2)u with 2d/(2d - 1) < y < 2, y ≥ α > 1. This equation is scaling-critical in Hsc with sc (y-α)/2. We show that the solution scatters in Hsc,1 where Hsc, 1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by. For this purpose we use the recently developed Strichartz estimate which is L2θ -averaged on the unit sphere Sd-1.

AB - In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index a between 1 and 2, and Hartree type nonlinearity F(u) = µ(|x|-y*|u|2)u with 2d/(2d - 1) < y < 2, y ≥ α > 1. This equation is scaling-critical in Hsc with sc (y-α)/2. We show that the solution scatters in Hsc,1 where Hsc, 1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by. For this purpose we use the recently developed Strichartz estimate which is L2θ -averaged on the unit sphere Sd-1.

KW - Angularly averaged strichartz estimate

KW - Hartree type fractional schrödinger equation

KW - Scaling critical space

KW - Small data scattering

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Small data scattering of hartree type fractional schrödinger equations in a scaling critical space

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