Interface motion and blow-up phenomena in nonlinear partial differential equations

非线性偏微分方程中的界面运动和爆炸现象

基本信息

批准号：
17340044
负责人：
MATANO Hiroshi
金额：
$ 10万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340044/
关键词：
nonlinear analysis partial differential equation blow-up of solutions interface motion asymptotic method singular limit nonlinear diffusion equations nonlinear heat equation 進行波

项目摘要

The aim of this research project is to make a theoretical study of various nonlinear problems related to interfacial motions and blow-up phenomena, by developing asymptotic methods based on the theory of infinite dimensional dynamical systems and stochastic methods, and also performing numerical simulations if necessary. We have obtained the following results.(1) We have given an optimal estimate concerning the singular limit of Allen-Cahn type nonlinear diffusion equations, that has not been known previously. We also obtained similar results for FitzHugh-Nagumo systems and Lotka-Volterra competition systems.(2) We have clarified the global dynamics of blow-up solutions in nonlinear diffusion equations. This was made possible by extending the existing theory on global attractors to the case where blow-up occurs.(3) In nonlinear heat equations with power nonlinearity, the blow-up of solutions is classified into type I and type II, the latter being much more difficult to analyze than the … More former. By using a topological method based on the braid group theory, we have succeeded in determining all possible type II blow-up rates.(4) We studied the stationary problem for the Allen-Cahn equation on the 2-dimensional space having lattice periodicity by using variational methods. We have shown that a necessary and sufficient condition for the existence of multi-layered stationary solution is that the set of single layered solutions has a gap somewhere ; in other words, this set should not be a continuum (foliation).(5) We considered periodic traveling waves in a two-dimensional infinite strip whose boundaries are saw-tooth shaped. We determined the homogenization limit of such traveling waves as one lets the boundary undulation finer and finer.(6) Various partial differential equations are derived from microscopic models via hydrodynamic limit. Those equations include the Stefan problem and some stochastic differential equations.(7) Regularity of free boundaries arising in various elliptic equations such as the combustion model has bee established.(8) Stability of V-shaped traveling wave in the equation of curvature-dependent motion has been established. Less

该研究项目的目的是通过基于无限尺寸动态系统和随机方法的理论来开发渐近方法，对与界面运动和爆炸现象相关的各种非线性问题进行理论研究，并在必要时执行数值模拟。我们已经获得了以下结果。（1）我们对Allen-CAHN型非线性扩散方程的奇异极限进行了最佳估计，这还没有获得针对Fitzhugh-Nagumo Systems和Lotka-Volterra Systems的类似结果。（2）我们已经确定了非线性扩散方程中爆破溶液的全球动力学。通过将现有关于全球吸引子的现有理论扩展到发生爆炸的情况。（3）在具有功率非线性的非线性热方程式中，解决方案的爆炸被归类为I型和II型，后者比…更难以分析。通过使用基于编织组理论的拓扑方法，我们成功地确定了所有可能的II型爆炸率。（4）我们在使用变异方法的两维空间上研究了艾伦-CAHN方程的固定问题。我们已经表明，多层固定解决方案存在的必要条件是，单个层次溶液的集合在某个地方具有差距。换句话说，这组不应是连续的（叶子）。（5）我们考虑了二维无限条带的周期性行进波，其边界是锯齿状的。我们确定了这样的行进波的均质化极限，例如使边界起伏越来越细。（6）各种偏微分方程通过微观模型通过流体动力学极限得出。这些方程式包括Stefan问题和一些随机微分方程。（7）在各种椭圆方程（例如组合模型）中产生的自由边界的规律性已建立。（8）已经确定了V形行进波的稳定性在曲率依赖性运动方程中的稳定性。较少的