Mathematical Open Problems of the Navier-Stokes Equations

纳维-斯托克斯方程的数学开放问题

• 批准号: 09304023
• 负责人: OKAMOTO Hisashi
• 金额: $ 13.57万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09304023/
• 关键词: Fast summation method for the vortex method; Kolmogorov flow; the Burgers equation; point vortex; internal iayer; bifurcation of surface wave; thermal convection; singularity of solutions; Navier-Stokes方程式; 水面波; 領域分割法; 渦構造; スペクトル法; Euler方程式 /

Progress is made in the study of the Navier-Stokes equation, the Burgers equations, and the reaction-diffusion equations. Okamoto and Shoji performed numerical experiments on the bifurcation of surface waves. New bifurcation diagrams are found and will be published in a form of textbook by World Scientific Inc. Okamoto and Sakajo compute numerically two-dimensional and three-dimensional vortex sheet motion. T.Ikeda and H.Ikeda consider a certain system of reaction-diffusion equations for three competing species. They clarify the structure of steady-states and traveling pulses. Their stability is also determined. K.Ohkitani and M.Yamada consider what is called the shell model of the turbulence. By numerical methods, they compute the Lyapunov numbers of the system and they study the scaling properties of the numbers. They derive an asymptotic formula as the viscosity tends to zero. T.Nakaki consider the motion of vortex patches as well as point vortices. He finds that the motion of the patches are quite similar to that of point vortices if the size of the patches are small enough and that the motion of vortex patches are substantially different if the sizes are large. Y.Kimura considers the motion of point vortices on two-dimensional hyperbolic surfaces. Its Hamiltonian formalism are derived and the algebraic properties of the invariants are studied. T.Nishida numerically computes the Boussinesq equations, which are the master equations for the thermal convection. In particular he obtains numerically the bifurcation from the trivial solution to the stationary convective flow. He applies the numerical verification technique and derives new criteria.

Navier-Stokes方程、Burgers方程和反应扩散方程的研究取得了进展。Okamoto和Shoji对表面波的分叉进行了数值实验。发现了新的分叉图，并将由世界科学公司以教科书的形式出版。Okamoto和Sakajo数值计算了二维和三维涡面运动。T.Ikeda和H.Ikeda考虑了一类三种群竞争的反应扩散方程组。它们阐明了稳态和行进脉冲的结构。它的稳定性也是确定的。K.Ohkitani和M.Yamada考虑所谓的湍流壳模型。通过数值方法，他们计算系统的李雅普诺夫数，并研究了这些数的标度特性。他们导出了当粘性趋于零时的渐近公式。T.Nakaki考虑了涡斑和点涡的运动。他发现，如果斑块的尺寸足够小，斑块的运动与点涡的运动非常相似，而如果斑块的尺寸很大，涡斑的运动则有很大的不同。木村认为运动的点涡二维双曲曲面。导出了它的哈密顿形式，并研究了其不变量的代数性质。T.Nishida数值计算Boussinesq方程，这是热对流的主方程。特别是他获得数值分岔平凡的解决方案，以固定的对流。他应用数值验证技术并推导出新的准则。

项目成果

H.Okamoto: "Exact solutions of the Navier-Stokes equations via Leray's scheme" Japan J.Indus.Appl.Math.14. 169-197 (1997)

H.Okamoto：“通过 Leray 方案精确求解纳维-斯托克斯方程”Japan J.Indus.Appl.Math.14。

M.Yamada: "Asymptotic formulae for the Lyapunov spectrum of fully-developed shell model turbulence" Phys.Rev.E.に掲載予定.

M.Yamada：“完全发展的壳模型湍流的李亚普诺夫谱的渐近公式” 计划发表在 Phys.Rev.E 上。

T.-P.Liu, A.Matsumura, and K.Nishihara: "Behaviors of solutions for the Burgers equation with boundary corresponding to rar-efaction waves" SIAM J.Math.Anal.29. 293-308 (1998)

T.-P.Liu、A.Matsumura 和 K.Nishihara：“边界对应于稀疏波的 Burgers 方程解的行为”SIAM J.Math.Anal.29。

H.Okamoto: "Exact solutions of the Navier-Stokes equations via Leray's scheme" Japan Journal of Industrial and Applied Mathematics. vol.14. 169-197 (1997)

H.Okamoto：“通过 Leray 方案精确求解纳维-斯托克斯方程”，《日本工业与应用数学杂志》。

H.Ikeda and T.Ikeda: "Bifurcation phenomena from standing pulse solutions in some reaction-diffusion systems" J.Dynamics and Differential Equations. to appear. (1999)

H.Ikeda 和 T.Ikeda：“某些反应扩散系统中驻留脉冲解的分岔现象”J.Dynamics and Differential Equations。