Mathematical Analysis of partial differential equations related to a variational problem via the discrete Morse Semiflows

我们主要通过离散莫尔斯半流研究了与变分问题相关的偏微分方程。我们主要关注解的奇异点集合。这样的设置有时需要大量的精力。因此，我们可以认为我们的目的是在解的奇点上处理能量集中现象。在这种观点下，我们处理了以下类型的问题：(1)开发一个解决最小化问题的平行机，(2)通过最小化过程开发一种数值方法，(3)开发一种通过最小化来解决抛物线和双曲方程的方法。针对这些问题，我们开发了一台8 cpu并行计算机来解决最小化问题。利用这一方法，我们对方程、金兹堡-朗道系统和液晶问题的奇异性结构进行了数值计算。抛物型和双曲型问题的离散莫尔斯半流基本方法。我们还解决了BBM-Burgers方程孤立波解的渐近性质。此外，我们还开发了一个求解双曲自由边界问题的软件。这是基于方程的平滑方法，即使自由边界改变了拓扑结构，我们也能得到很好的结果。我们将这些结果总结为8篇论文（已发表或已出版）和2篇预印本（已提交）。

We mainly investigated partial differential equations related to a variational problem via the discrete Morse semiflows. Our main interest is on sets of singular points of a solutions. Such sets has sometimes big energy concentrate on it. So, we can cosider that our purpose is on treating the energy concentration phenomena on the singularity of solutions. In this stand point of view, we treated the following type of problems :(1) Develop a prallel machine for solving mininizing problems,(2) Develop a Numerical method via a minimization process,(3) Develop a method to solve both parabolic and hyperbolic equations via minimizing.For these problems, we have developped a 8-CPU parallel computer for solving minimizing problems. By use of this, we did a numerical copmutations to catch the structure of singularities for eikonal equation, Ginzburg-Landau system, and smestics liquid crystal problems. Basic method due to discrete Morse semiflow for parabolic and hyperbolic problems.We also solved the asymptotic behavior of solitary wave solutions for BBM-Burgers equations.Moreover we developped a software to solve hyperbolic free boundary problems. This is based on the smoothing method of a equation and we can get good results even when the free boundary changes its topology.We summed up these results into 8 papers (appeared or in press) and 2 preprint (submitted).