Studies on singular perturbation problems in nonlinear mechanics

非线性力学奇异摄动问题的研究

• 批准号: 11304005
• 负责人: OKAMOTO Hisashi
• 金额: $ 10.66万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11304005/
• 关键词: singularity; blow-up of solutions; interior layer; self-similar solution; bifurcation; dynamical system; periodic solution; Navier-Stokes equations!; ナビエストークス方程式; ボルツマン方程式; 衝撃波; KdV方程式; 水面波; 渦力学; 反応拡散系

(1) New phenomena on the Navier-Stokes equations were found. Among others, solutions having interior layers and those solutions having k-10 spectra are remarkable. (2) Bifurcation phenomena in surface waves were clarified. In particular, an accurate numerical method was developed for singular solitary waves. (3) dynamical systems viewpoints on the shell model of turbulence proposed by Ohkitani and Yamada were enhanced. (4) applications to reaction-diffusion systems, (5) vortex formation in the 2-dimensional decaying turbulence by Y. Kimura. (6) asymptotic behavior of shock wave solutions was clarified by Kawashima and Matsumura.Okamoto, with the aid by Kim Sunchul, analyzed the bifurcating solutions arising in the rhombic periodic flows. It was demonstrated, by an elaborate numerical computations, that some solutions have k-10 spectra as the Reynolds number tends to infinity. Okamoto and A. Craik considered a three-dimensional dynamical system arising in fluid mechanics. Two different … More solutions, one with 90-degree bending and one without bending, were found and the mechanism of them was theoretically explained.Y. Kimura, with J. Herring, successfully explained theoretical background of vortex structures arising in rotating fluid. S. Kawashima proved the well-posedness of radiating gases.T. Ikeda considered models for combustion synthesis. With numerical experiments he demonstrated that the solutions of the model can reproduce the results of the laboratory experiments.H. Ikeda and H. Okamoto considered a special solution of the Navier-Stokes equations called Oseen flows. Some interior layers was rigorously proved. H. Ikeda also proved that a Hopf bifurcation occurs in the traveling wave solutions of a certain bi-stable system of reaction diffusion.H. Fujita proved the existence of the solutions of the Navier-Stokes equations when they are subjected to a leak boundary condition. He also derived a new convergence rate of the domain-decomposition method.M. Yamada and K. Ohkitani discovered, by a numerical experiments, a time-periodic solution, which simulate the turbulent motions of real flows. Less

（1）发现Navier-Stokes方程的新现象。除其他外，具有内部层的解决方案和具有K-10光谱的解决方案非常出色。 （2）阐明了表面波中的分叉现象。特别是，为单数固体波开发了一种准确的数值方法。 （3）Ohkitani和Yamada提出的湍流模型上的动态系统观点得到了增强。 （4）对反应扩散系统的应用，（5）Y. Kimura在二维衰减湍流中的涡流形成。 （6）川岛和松村阐明了冲击波溶液的不对称行为。冈本在金·森彻尔（Kim Sunchul）的帮助下分析了龙骨周期性流中产生的分叉溶液。通过精心制作的数值计算证明，某些解决方案具有K-10光谱，因为雷诺数趋向于无穷大。冈本和A. craik考虑了在流体机制中产生的三维动态系统。发现了两种不同的……更多的解决方案，一种具有90度弯曲和一个没有弯曲的解决方案，理论上解释了它们的机制。 Kimura与J. Herring一起成功地解释了旋转流体中产生的涡旋结构的理论背景。 S. kawashima证明了辐射气体的适当性。 Ikeda考虑了合成组合的模型。通过数值实验，他证明该模型的解决方案可以重现实验室实验的结果。 Ikeda和H. Okamoto考虑了称为Oseen Flow的Navier-Stokes方程的特殊解决方案。严格证明了一些内部层。 Ikeda还证明，在某种双稳定反应扩散系统的行驶波解决方案中发生了HOPF分叉。藤田证明了在遇到泄漏边界条件时，Navier-Stokes方程的解决方案的存在。他还得出了域分解方法的新收敛速率。 Yamada和K. Ohkitani通过数值实验发现了一个时间周期性解决方案，该解决方案模拟了真实流的湍流运动。较少的

项目成果

H.Okamoto, M.Shoji: "World Scientific Publ."A Mathematical Introduction to Permanent Periodic Progressive Waves. 228 (2001)

H.Okamoto、M.Shoji：“世界科学出版社”永久周期行进波的数学简介。

H.Okamoto and X.Chen: "Global Existence of Solutions to the Proudman-Johnson Equation"Proc.Japan Acad.. 76. 149-152 (2000)

H.Okamoto 和 X.Chen：“Proudman-Johnson 方程解的全局存在性”Proc.Japan Acad.. 76. 149-152 (2000)

H.Ikeda and T.Ikeda: "Bifurcation phenomina from standing pulse solutions of bistable reaction-diffusion systems"J. Dynamics and Differential Equations (to appear). (2000)

H.Ikeda 和 T.Ikeda：“双稳态反应扩散系统的固定脉冲溶液的分叉现象”J。

M.Nagayama, H.Okamoto, J.Zhu: "On the blow-up of some similarity solutions of the Navier-Stokes equations"to appear in Quader.di Mat..

M.Nagayama、H.Okamoto、J.Zhu：“关于纳维-斯托克斯方程的一些相似解的放大”出现在 Quader.di Mat..

M.Nagayama, H.Okamoto: "On the interior layer appearing in the similarity solutions of the Navier-Stokes equations"to appear in Japan J.Indust.Appl.Math..

M.Nagayama、H.Okamoto：“论纳维-斯托克斯方程相似解中出现的内层”发表于日本 J.Indust.Appl.Math..