Research on the time global solutions for the partial differential equations of Kirchhoff type

基尔霍夫型偏微分方程时间全局解的研究

• 批准号: 14540188
• 负责人: YAMAZAKI Taeko
• 金额: $ 1.09万
• 依托单位: Tokyo University of Science
• 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-14540188/
• 关键词: Kirchhoff equations; global solvability; hyperbolic equations; dissipative term; forcing term; spectral decomposition; linear wave equations; decay estimate

1.We studied the initial value problem for the dissipative hyperbolic equation of Kirchhoff type associated to a non-negative self-adjoint operator A in a Hilbert space, with a dissipative term and a forcing term.We gave sufficient conditions on the forcing term, related with the spectral decompo-sition of the operator A, for the global unique solvability of this initial value problem with small inital value.2.We studied the unique global solvability of initial(boundary)value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three.The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case ; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. We gave sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation.

1.我们研究了与非阴性自我接合接合操作员A相关的Kirchhoff类型的耗散双曲线方程的初始值问题，其在Hilbert空间中具有耗散项和强迫术语。我们在强迫术语中给出了足够的条件，与操作员A的频谱分解组合有关，对于此初始值问题的全局唯一可溶解性，具有小的inital值。2。我们研究了外部域或Kirchhoff方程的初始（边界）值问题的唯一全局可溶解性或外部域或在整个欧几里得空间中，尺寸大于三个。在某些加权Sobolev空间中，对于外部域情况，初始数据的广义傅立叶变换足够小。我们通过表明该方程的全局溶解度从线性波方程解的时间衰减估计值遵循的是，我们就通常的初始数据的Sobolev规范提供了足够的条件。

项目成果

Taeko Yamazaki: "Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three"Mathematical Methods in the Applied Sciences. (in press).

Taeko Yamazaki：“基尔霍夫方程在维度大于三的外部域中的全局可解性”应用科学中的数学方法。

Naoyuki Ishimura, Takeo K.Ushijima: "An elementary approach to the analysis of exact solutions for the Navier-Stokes stagnation flows with slips"Archiv der Mathematik. (発表予定).

Naoyuki Ishimura、Takeo K.Ushijima：“分析带有滑移的纳维-斯托克斯停滞流精确解的基本方法”Archiv der Mathematik（待提交）。

Taeko Yamazaki: "Sufficient conditions on the forcing terms for the global solvability of dissipative Kirchhoff equations"Nonlinear Analysis. 51-6. 969-982 (2002)

Taeko Yamazaki：“耗散基尔霍夫方程全局可解性的强迫项的充分条件”非线性分析。

Reido Kobayashi, Masayuki Tamura: "Soliton solution of the KdV equation and Crum's theorem"Advances in Mathematical Sciences and Applications. 12・2. 779-784 (2002)

Reido Kobayashi、Masayuki Tamura：“KdV 方程的孤子解和 Crum 定理”数学科学与应用进展 12・2（2002 年）。

Kazumine Moriyasu, Masatoshi Oka: "Differentiable maps having the informly shadowing property"Topology and its applications. 122-(1-2). 377-396 (2002)

Kazumine Moriyasu、Masatoshi Oka：“具有非正式阴影特性的微分映射”拓扑及其应用。