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

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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]).