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

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支曲线，我们采用了解析法结合计算分析和计算机辅助证明的方法。作为计算机辅助证明，我们提出了与参数值相对应的解的存在性判据。利用该方法，我们证明了对应于大Rayleigh数的整体分支曲线的存在性，在三维情形下，我们数值研究了Roll型、矩形和六边形解的图形形成及其稳定性，并阐明了局部分支理论中看不到的整体分支图.泰勒问题：考虑了两圆柱反向转动时库埃特流的稳定性。将其化为常微分方程组的特征值问题， 关于我们 方程的解，并通过计算机辅助证明，得到了发生稳定分岔或Hopf分岔的临界泰勒数。具有重数的分歧点是我们未来的研究课题之一.利用数值验证方法证明了Navier-Stokes方程定常解的存在性定理，至少在小的代数数下.动力学系统：我们知道当向量场奇点的简并度增加时，动力学行为变得更加复杂，全局现象变得更加包含在内。研究了余维为3的奇异点，并解析地证明了异宿环的分岔和混沌吸引子的分岔.对于三维定常Navier-Stokes方程的外问题，引入Morrey空间的真实的插值来求解N-S方程，成功地构造了外定常解，并证明了其稳定性.少

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