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和处理其中的全局分叉曲线，我们采用了计算分析和计算机辅助证明相结合的分析方法。作为计算机辅助证明，我们提出了与参数值对应的解的存在性的证明准则。利用这种方法，我们证明了存在对应于大Rayleigh数的滚动型解的全局分叉曲线的存在性；在三维情形下，我们数值研究了滚动型、矩形和六角型解的模式形成及其稳定性，并澄清了从局部分叉理论中看不到的全局分岔图。泰勒问题：我们考虑了两个圆柱体反向旋转时Couette流的稳定性。将其归结为…常微分方程组的特征值问题更多的方程和它可以通过我们的计算机辅助证明，以确定确切的临界泰勒数，在那里发生平稳或霍普夫分叉。具有多重性的分岔点是我们未来的研究课题之一。用我们的数值验证方法证明了N-S方程定常解的存在性，至少对于小的雷诺数是成立的。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，全局现象变得更具包容性。我们研究了余维为3的奇点，并解析地证明了异宿环分支和混沌吸引子分支。对于定常Navier-Stokes方程的三维外问题，我们引入了Morrey空间的实内插法来求解N-S方程，在没有非自然零净力条件的情况下，成功地构造了定常外问题的解，并证明了它的稳定性。较少

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