非線型分散型方程式の大域解の存在とその漸近挙動についての研究

非线性分布方程全局解的存在性及其渐近行为研究

• 批准号: 03J09736
• 负责人: 山崎 泰子
• 金额: $ 1.15万
• 依托单位: Hokkaido University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03J09736/
• 关键词: 非線型分散型方程式; 非線型消散型方程式; 解の解析性; 解の分散性; 数理物理; 函数解析; 古典場の理論; 散乱理論

非線型分散型消散型の初期値問題の時間大域な可解性及び散乱問題,特に解の漸近挙動の記述についての研究を目的とする.これらの問題は,非線型分散型消散型方程式の時間大域解の新たな函数空間のクラス及び非線型項,分散項,消散項の指数と臨界指数を与え,より精密な解の漸近挙動を記述すると云う目的の上で非常に重要である.2000年以来私はSchrodinger(NLS)方程式(非線型分散型方程式)を中心に行ってきた.また1970年の後期より『分散項』を含んだ非線型分散型方程式の時間大域解に関する研究は盛んに行われている.その中の一方程式である私の取り組んでいるNLS方程式ついては非線型項の指数と空間次元との間に密接な関係が有り,特に長距離の散乱理論の立場からNLS方程式の非線型項をL^2空間の枠組みでSupercritical, Critical, Subcriticalの3つの場合に分類し特徴付けが出来る.特に,今年度は非線型項に積分項を考え,上述の分類によればCriticalに匹敵する小澤氏と山内氏との共著で解の解析性について書き上げた.また,申請者はSupercriticalな『分散項』の指数が2以上3以下である論文を完成した.このように『枠組み』は,NLS型方程式だけではなく他の種類の方程式にも適用可能と思われるが,現在構築段階にあり完全な状況ではない.申請者はこの『枠組み』を一部にすぎないがこれを構築した.更に,『消散項』をこれらの式に付随して考えた.特にCritical, SubcriticalなNLS方程式に消散項を加えた場合を考えた.このように,非線型分散型消散型方程式に於ける大域解の存在及び散乱理論に関する研究並びにこれらの研究結果に『消散項』を伴う非線型の取る指数の正確な範囲を明確に,より精密な漸近挙動を表現する事である.

In order to solve the problem of large-scale time domain solution and dispersion in the initial stage of non-linear dispersive operation, the purpose of this study is to solve the problem of recent movement record and research purpose. Non-linear dispersive equations, non-linear dispersive equations, time domain solutions, new functions, spatial equations and non-linear terms, dispersive terms, dissipative item index boundary index, accurate solution of recent motion records are very important. Since 2000, the center of the private Schrodinger (NLS) equation (non-linear decentralized equation) has been very important. In the later period of 1970, the "dispersion term" included the time domain solution of non-linear decentralized equations. In this paper, we use the one equation to obtain data from the NLS equation. The index of the non-linear item is not valid. There is a close connection between the non-linear items and the non-linear equation. In the NLS equation, the L2-type spatial data sets such as Supercritical, Critical, Subcritical 3 are available. In particular, this year's non-linear projects are actively divided into projects. The above-mentioned categories are similar to those of Yamauchi Yamauchi. They are the co-authors of analytical research. Applicants are required to complete the Supercritical documents with an index of more than 2 and less than 3. It is possible to use the NLS equation to solve the problem of other equations, but now that the equation is in full condition, it is possible to use the NLS equation. The applicant is responsible for the registration of one of the applicants. What's more, the dissipative items will be paid in the same way as the dissipative items. Special Critical, Subcritical's NLS equation dissipates terms and adds data to the test. Non-linear dispersive equations are widely used in the study of existence and dispersion theory. The results show that the "dissipative term" is associated with the correct range of non-linear data acquisition index.

项目成果

Analytic smoothing effect for solutions to Schrodinger equations with nonlinearity of integral type

积分型非线性薛定谔方程解的解析平滑效应

• 发表时间: 2005
• 期刊: Osaka Journal of Mathematics 42
• 作者: S.Ding;N.Murakami;H.Tomiyama;H.Takada;Y.Cho;Y.Cho;Y.Cho;Y.Cho;Y.Cho;Cho Yonggeun;Cho Yonggeun;Cho Yonggeun;Tohru Ozawa;Cho Yonggeun;Tohru Ozawa
• 通讯作者: Tohru Ozawa

山崎 泰子: "Life-span of smooth solutions to the complex Ginzburg-Landau type equation on a torus"Nonlinearity. 16. 2029-2034 (2003)

Yasuko Yamazaki：“圆环上复杂的 Ginzburg-Landau 型方程的光滑解的寿命”16。2029-2034 (2003)。

山崎 泰子: "Smoothing effect and large time behavior of solutions on NLS with nonlinearity of integral type"Communications in contemporary Mathematics. (発表予定).

Yasuko Yamazaki：“积分型非线性 NLS 解的平滑效应和大时间行为”当代数学通讯（待提交）。