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)发现了Navier-Stokes方程的新现象。其中，具有内层的溶液和具有k-10光谱的溶液是显著的。(2)澄清了表面波的分岔现象。特别是对奇异孤立波，提出了一种精确的数值计算方法。(3)强化了Ohkitani和Yamada关于湍流壳模型的动力学系统观点。(4)在反应扩散系统中的应用；(5)木村Y.在二维衰减湍流中的涡形成。(6) Kawashima和Matsumura澄清了激波解的渐近性质。冈本在金顺哲的帮助下，分析了在菱形周期流中产生的分岔解。通过详细的数值计算，证明了当雷诺数趋于无穷大时，某些解具有k-10谱。冈本和a .克雷克考虑了流体力学中出现的三维动力系统。找到了两种不同的解，一种是90度弯曲解，另一种是不弯曲解，并从理论上解释了它们的机理。木村和J. Herring成功地解释了旋转流体中涡结构产生的理论背景。川岛证明了辐射气体的适定性。池田考虑了燃烧合成的模型。通过数值实验，他证明了该模型的解可以再现实验室实验的结果。Ikeda和H. Okamoto考虑了Navier-Stokes方程的一种特殊解，称为Oseen流。对一些内层进行了严格的证明。H. Ikeda还证明了某双稳定反应扩散系统的行波解存在Hopf分岔。Fujita证明了在泄漏边界条件下Navier-Stokes方程解的存在性。他还推导出一种新的收敛速度的区域分解方法。Yamada和K. Ohkitani通过数值实验发现了一个时间周期解，它模拟了真实流动的湍流运动。少

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.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。

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)

Y. Kimura: "Vortex Motion on surfaces with constant curvature"Proc. R. Soc. London Ser A. 455. 245-259 (1999)

Y. Kimura：“具有恒定曲率的表面上的涡运动”Proc。

S. Kawashima and S. Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics"Indiana Univ. Math. J.. 50. 567-589 (2001)

S. Kawashima 和 S. Nishibata：“辐射流体动力学中双曲椭圆耦合系统的奇异极限”印第安纳大学。