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2023 Fiscal Year Final Research Report

Research on global analysis and concentration energy for nonlinear dispersive equations

Research Project

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Project/Area Number 18H01129
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionKobe University

Principal Investigator

Takaoka Hideo  神戸大学, 理学研究科, 教授 (10322794)

Project Period (FY) 2018-04-01 – 2023-03-31
Keywords分散型方程式 / 初期値問題 / 適切性 / 大域挙動
Outline of Final Research Achievements

I developed the global existence and large time behavior of solutions to nonlinear dispersive equations, especially the nonlinear Schrodinger equation and KdV equation, which describe the wave phenomena produced by the incorporation of nonlinear and dispersive effects. Regarding the nonlinear Schrodinger equation, I showed a phenomenon in which the wave energy of the solution is converted due to the concentration effect of the solutions due to the resonance/non-resonance in the interaction for nonlinear terms. Regarding the KdV equation, by considering the bilinear estimates, I showed the smoothing effect of the KdV equation and the global solvability of the Zakharov-Kuznetsov equation, which is a two-dimensional extension of the KdV equation.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

ハミルトニアンに対応したエネルギー保存空間は,ハミルトニアンが意味をもつ空間として意味があり,その関数空間における解の大域存在,大域挙動に関する研究は多い.また,弱解の一意存在性や正則性に関する解析では,方程式に対してスケール不変である関数空間が重要な役割を果たし,そのような関数空間はエネルギー保存量が有限とは限らない場合が多い.本研究の一つ目の問いは,初期値問題の適切性・非適切性を切り分ける臨界空間はどうかということである.二つ目の問いは,解の大域的な振る舞いをフーリエ空間における波動のエネルギー密度の転換過程で解析できないかということであった.本研究で得られた成果の価値は高いと言える.

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Published: 2025-01-30  

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