Experimental, numerical and mathematical approach to the complexity of the phenomena of interfaces

在这个项目中，我们关注以下现象。1)光敏Belousov-Zhabotinsky反应中螺旋波的模式形成和自传播高温合成中的螺旋波。2)细菌细胞群的自组织集落模式。3)聚合物溶液中粘性指指和气泡运动的Hele-Shaw细胞实验。4)多孔介质流动中扩散与吸收相互作用引起的支撑分裂现象。5)断裂力学中裂纹路径的预测与分析6)晶体生长问题中的几何模型。我们得到了以下结果。在1)和2)中，用反应扩散方程描述了图案形成的机理，并利用奇异极限方法和分岔理论（[1],[2],[3],[4],[5]）对其进行了分析。3)考虑剪切减薄指数，推导出修正的达西定律，对手指速度（[6],[7]）有较好的预测。描述这种现象的数学模型被写成具有强吸收的非线性扩散方程。得到了支撑开始分裂为两个不相交集的充分条件（[8],[9]）。推导了光滑裂纹方向与曲率的关系式。这些公式的有效性在简单的例子（[10],[11]）中得到了证明。6)提出了基于能量不可微梯度系统的面形晶体生长模型和晶界模型。获得了证明和分析这些模型的数学基础，理论和数值方法显示了这些模型的解是如何演变的（[12],[13],[14]）。

In this project we are concerned with the following phenomena.1) Pattern formation of spiral waves in the photosensitive Belousov-Zhabotinsky reaction and helical waves in self-propagating high-temperature syntheses.2) Self-organized colony patterns by a bacterial cell population.3) Hele-Shaw cell experiments of viscous fingering and bubble motion in polymeric solutions.4) Support splitting phenomena caused by the interaction between diffusion and absorption in porous medium flow.5) Prediction and analysis for the crack path in fracture mechanics6) Geometric models in crystal growth problems.We have obtained the following results.In 1) and 2) the mechanism of the pattern formation is described by the reaction-diffusion equations and are analyzed by using the method of the singular limit and the theory of bifurcation ([1], [2] , [3] , [4] , [5]).In 3) The modified Darcy's law is derived by taking account of the index of shear-thinning, and gives a good prediction of the finger velocity ([6], [7]).In 4) the mathematical models which describe such phenomena are written as the nonlinear diffusion equation with strong absorption. The sufficient conditions under which the support begins to split into two disjoint sets are obtained ([8], [9]).In 5) some formulas connecting the direction and the curvature of the smooth cracks are derived. The validity of these formulas are shown in simple examples ([10], [11]).In 6) Models of faceted crystal growth and of grain boundaries are proposed based on the gradient system with nondifferentiable energy. The mathematical basis for justifying and analyzing these models is obtained, and theoretical and numerical approaches show how the solutions of these models evolve ([12], [13], [14]).