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Analysis on Nonlinear Partial Differential Equations

非线性偏微分方程分析

基本信息

批准号：
05452009
负责人：
GIGA Yoshikazu
金额：
$ 4.03万
依托单位：
Hokkaido University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1993
资助国家：
日本
起止时间：
1993 至 1995
项目状态：
已结题

项目摘要

Motion of crystal surface in crystal growth is a typical example of phase-boundaries (interface). Such a phenomena attracts interdeciplinary interest as nonequilibriun nonlinear phenomena. Interface controlled model is an important class of evolution equations of phase boundaries. This is the case when heat and mass diffusion is negligible so that the evolution is determined by geometry of surface. Phenomena that facets appear on interface arises, for example, in the growth of Helium crystal growth in low temperature. In this situation, the governing equation has a nonlocal term and it is difficult to describe. So far the evolution law is described by restricting a class of evolving interfaces. The head investigator gave a formulation to this problem which is comparible with partial differential equations. It is based on the theory of nonlinear semigroups and nowadays it is called Fukui-Giga formulation. By this formulation curve evolution by crystalline energy can be understood as a limit of evolution by smooth anisotropic energy.In motion of interfacial energy having anisotropy, it is important whether or not there is a self-similar shrinking solution. If interfacial energy is isotropic and there is no external force, the equation becomes the famous curve shortening equation. It is known that the only self-similar solution is a circle. However, the proof is rather complicated. Head investigator gave an elementary proof. For motion by anisotropic curvature be proved the existence of self-similar solution in an elementary way. However, uniqueness is shown only for evolution law that does not depend the orientation of curves.The above research is a study of important example of nonlinear parabolic equations.Investigator studied large time asymptotic behaviors of solutions of nonlinear Schrodinger equation describing dispersive phenomena and discovered a nonlinear effect that is not tractable as a linear phenomena.

晶体生长过程中晶体表面的运动是相界（界面）的一个典型例子。这种现象作为非平衡非线性现象引起了十余年的研究兴趣。界面控制模型是一类重要的相界演化方程。这是当热量和质量扩散可以忽略不计时的情况，因此演化由表面的几何形状决定。例如，在低温下的氦晶体生长中，界面上出现刻面的现象。在这种情况下，控制方程具有非局部项，很难描述。到目前为止，演化规律是通过限制一类演化界面来描述的。首席研究员给了这个问题的一个公式，这是比较偏微分方程。它基于非线性半群理论，现在被称为Fukui-Giga公式。根据该公式，由结晶能引起的曲线演化可以理解为由光滑各向异性能引起的演化的极限。在具有各向异性的界面能运动中，是否存在自相似收缩解是重要的。如果界面能是各向同性的，并且没有外力，则该方程成为著名的曲线缩短方程。已知唯一的自相似解是圆。然而，证明是相当复杂的。首席调查员给出了初步的证据。对于各向异性曲率运动，用初等方法证明了自相似解的存在性。然而，唯一性只表现在不依赖于曲线方向的演化规律上。上述研究是对非线性抛物型方程的重要例子的研究，研究者研究了描述色散现象的非线性薛定谔方程解的大时间渐近行为，发现了一种非线性效应，这种非线性效应不像线性现象那样容易处理。