Comprehensive Research Toward the Global Theory for the System of Nonlinear Partial Differential Equations

非线性偏微分方程组整体理论的综合研究

• 批准号: 10304012
• 负责人: NISHIDA Takaaki
• 金额: $ 17.66万
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10304012/
• 关键词: Nonlinear PDE; Global bifurcation structure; Bifurcation from degenerate singular points; Dynamical system; Computer assisted proof; Navier-Stokes equation; Heat convection problem; Taylor-Couette problem; 縮約系の大域的解析; 精度保証付き数値計算; 粘性的衝撃波; 燃焼合成反応

1. Heat convection problem :In order to investigate the global structure of the solution space of the nonlinear PDE's and to treat the global bifurcation curves in it, we worked on the analytical method combined with the computational analysis and computer assisted proof. We proposed criterions to prove the existence of solutions which correspond to parameter values as computer assited proof. Using the method we showed the existence of global bifurcation curves on which the roll-type solutions exist that correspond to large Rayleigh numbers.In the case of 3-dimension we investigated numerically the pattern formation of roll-type, rectangle-typpe and hexagonaltype solutions and their stability, and we clarified the global bifurcation diagram which is not seen from the local bifurcation theory.2. Taylor problem :We considered the stability of Couette flow when the two cylinder rotate in the opposite directions. It is reduced to the eigenvalue problem for the system of ordinary differenti … More al equations and it can be treated by our computer assisted proof to see the exact critical Taylor number, at which the stationary or Hopf bifurcation occurs. The bifurcation point with multiplicity is one of our future subject.3. The existence theorem for stationary solution of Navier-Stokes equation is proved by our numerical verification method at least for small Reynols number.4. Dynamical systems :We know that when the degeneracy of singular points of vector field increases, the behavior of dynamics becomes more complex and the global phenomena become more included. We investigated the singular point with codimension 3 and proved analytically that the hetero-clinic cycle bifurcates and also chaotic attractor does.5. For the 3-dimensional exterior problem of stationary Navier-Stokes equation, we introduced a real interpolation of Morrey spaces to solve N-S equation and succeeded to construct the exterior stationary solution and to prove its stability without the unnatural zero net force conditions. Less

1.热对流问题：为了研究非线性偏微分方程解空间的全局结构并处理其中的全局分岔曲线，我们研究了计算分析和计算机辅助证明相结合的解析方法，提出了判据。为了证明与参数值相对应的解的存在性作为计算机辅助证明，我们使用该方法证明了存在与大瑞利数相对应的滚动型解的全局分岔曲线的存在性。三维数值研究了卷型、矩形型和六角型解的模式形成及其稳定性，并阐明了局部分岔理论中看不到的全局分岔图。 2．当两个圆柱体反向旋转时的库埃特流的问题被简化为常微分方程组的特征值问题，并且可以通过我们的计算机辅助证明来处理以查看精确的问题。临界泰勒数，其中出现平稳或Hopf分岔的点是我们未来的课题之一。 3．至少对于小雷诺数的数值验证方法证明了Navier-Stokes方程平稳解的存在性定理。 number.4. 动力系统：我们知道，当矢量场奇异点的简并性增加时，动力学行为变得更加复杂，并且全局现象变得更加包容。我们用余维来研究奇异点。 3．分析证明异宿循环分叉，混沌吸引子也分叉。5．针对平稳Navier-Stokes方程的3维外问题，引入Morrey空间的实数插值来求解N-S方程，并成功构造了外部固定解并在不自然的零净力条件下证明其稳定性。