Analysis of the scattering operator for nonlinear dispersive equations

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800074
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Chiba University
• Principal Investigator: Hironobu Sasaki 千葉大学, 大学院理学研究科, 准教授 (00568496)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 非線型分散型方程式 / 波動作用素 / 散乱作用素 / 解析的平滑化効果 / 解の漸近挙動 / ソボレフ空間 / ローレンツ空間 / ベゾフ空間

Outline of Final Research Achievements

In this study, we consider some problems on scattering operators and analytic smoothing effects for nonlinear dispersive equations. Using the Method of functional analysis, we obtain the following results:
(1) We considered the one space dimensional Dirac equation with a p-th power nonlinearity, and we proved that if p is greater than 3 then the scattering operator can be defined on a neighborhood of 0 in a suitable Hilbert space. (2) We considered the three space dimensional Dirac equation with a cubic power nonlinearity, and we proved that a neighborhood of 0 in the weighted Sobolev space is included in the same weighted Sobolev space. (3) We considered the Hartee equation, and we proved some properties of the analytic smoothing effect.

Free Research Field

偏微分方程式