Study on non-liner partial differential equations by means of besov tpe norms

基于besov tpe范数的非线性偏微分方程研究

• 批准号: 14540186
• 负责人: MURAMATSU Toshinobu
• 金额: $ 1.15万
• 依托单位: CHUO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540186/
• 关键词: Besov type norm; Fourier restiction norm; semilinear Schrodunger equation; trilinear estimates; KdV equtatiion; initial value problem; well-posednessT; 適切性; 三重線型評価; 非線型シュレディガー方程式; Besov空間; 臨界指数

We defined the Besov type norms which are generalizations of the Fourier restriction norm due to Bourgain, appilled them to the initial value problem of nonlimear partial differential equations, and obtained the following results :1. The intial value problem for the semilinear Schrodinger equation.(1)Quadratic nonlimearity case.For the case whetre the space dimension is 1 or 2 we proved that the intial value problem in the Sobolev space of the critical order is well-posed.We also obtained the results for the case where the space dimension is greater than 2. The key method is bilinear estimates by meas of Besov type norms.(2)Cubic nonlinearity caseWe proved that the initial value problem is well-posed in the Besov space of critical order when the space is 1, and this result is better than that obtained by the Fourier restriction norm. The key method is trilinear estimates by means of Besov type norms.(3) We find that the initial value problem is well-posed in the space of square integrable functions when the nonliniarity is the derivative of the squrer of the complex conjugate of the unknown function.2. The initial value problem for KdV equation.We proves that the initial value problem is well-posed in the Sobolev space which is very closed to that of order -3/4 (the critical order).

我们定义了BESOV类型规范，这些规范是由于波尔加恩而导致的傅立叶限制规范的概括，将它们粘贴到非限制偏微分方程的初始值问题中，并获得了以下结果：1。半连接schrodinger方程的INTIAL值问题。（1）二次非泄露情况。对于案例，空间维度为1或2，我们证明了关键顺序的Sobolev空间中的Intial值问题。获得了空间维度大于2的情况的结果。关键方法是通过测量BESOV类型规范的双线性估算。（2）立方非线性case我们证明了初始值问题在关键的BESOV空间中得到了很好的选择。当空间为1时，顺序比傅立叶限制规范获得的结果要好。关键方法是通过BESOV类型规范进行三连线估算。（3）我们发现，当非界限是未知的复杂偶联物的squrer的衍生物时，初始值问题在正方形集成函数的空间中得到了很好的范围。功能2。 KDV方程的初始值问题。我们证明，在Sobolev空间中，初始值问题在sobolev空间中占据了很好的范围，该空间非常接近-3/4（关键顺序）。

项目成果

田岡志婦: "Well-posedeness of the Cauchy problem for the semilinear Schro"dinger equation with qudratic nonlinear in Besov spaces"Hokkaido Mathematical J.. 掲載決定. (2004)

陶卡师傅：“半线性施罗的柯西问题的适定性”Besov空间中二次非线性的丁格方程”北海道数学杂志。接受出版。（2004年）

田岡志婦: "Well-posedness of the Cauchy problem for the semilinear Schro"dinger equation with qudratic nonlinearity in Besov spaces"Hokkaido Mathematical J.. 掲載決定.

陶卡师傅：“半线性施罗的柯西问题的适定性”Besov空间中二次非线性的丁格方程”《北海道数学杂志》接受发表。

Mochizuki, K., Trooshin, I.: "Inverse problem for interior spectral data of the Dirac opetators on a fimite interval"Publ. RIMS, Kyoto Univ.. Vol.38. 387-395 (2002)

Mochizuki, K., Trooshin, I.：“有限间隔上狄拉克算子的内部光谱数据的反演问题”Publ。

村松壽延: "The initial value problem for the 1-D semilinear Schro"dinger equation in Besov spaces"Journal of the Mathematical Society of Japan. 掲載決定. (2004)

Toshinobu Muramatsu：“Besov 空间中的 1-D 半线性 Schro”dinger 方程的初始值问题”日本数学会杂志。接受出版。（2004 年）

大春慎之助: "On the semigroups approach to age-dependent spatially distributed two sex models of population dynamics"Adv.Math.Sci.Appl.. 13. 423-442 (2003)

Shinnosuke Oharu：“关于人口动态的年龄依赖性空间分布两种性别模型的半群方法”Adv.Math.Sci.Appl.. 13. 423-442 (2003)