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

我们主要通过离散的Morse半流来研究与变分问题有关的偏微分方程。我们的主要兴趣是关于解的奇点集。这样的布景有时会有很大的能量集中在上面。因此，我们可以认为，我们的目的是处理解的奇异性上的能量集中现象。在这种观点下，我们处理了以下类型的问题：(1)开发了求解极小化问题的并行机；(2)通过极小化过程发展了一种数值方法；(3)发展了一种通过极小化求解抛物型和双曲型方程的方法。针对这些问题，我们开发了一台求解极小化问题的8 CPU并行计算机。利用这一点，我们进行了数值共变换，以捕捉爱柯尼亚方程、Ginzburg-Landau系统和Smestics液晶问题的奇点结构。基于离散Morse半流的基本方法求解抛物型和双曲型问题。我们还解决了BBM-Burgers型方程孤立波解的渐近行为。此外，我们还开发了求解双曲自由边界问题的软件。这是基于方程的光滑化方法，即使在自由边界发生拓扑变化的情况下也能得到很好的结果。我们将这些结果总结为8篇论文(已发表或正在出版)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).