Mathematical Analysis on Change of Patterns by Anisotropy and Diffusion

为研究非线性扩散方程由于各向异性和扩散效应引起解的形状变化的机理建立了数学基础。我们只给出两个典型的主题。（i）描述晶体表面运动的方程。当表面能各向异性较强时，如雪晶，在平衡状态下会出现一个称为小平面的平面。控制其演化的方程在形式上是一个曲率流方程。然而，扩散是如此之强，它不能被视为一个偏微分方程。研究了一个描述雪晶生长的Stefan型自由边界问题，假设雪晶的平衡形状为圆柱体。我们建立了（a）在小平面保持为小平面的假设下的局部可解性;（B）小平面附近扩散场行为的贝格效应;（c）小平面弯曲的充分必要条件;（d）小平面弯曲存在自相似解的充分条件和尺寸条件;（e）小平面保持在平衡附近的严格证明.另一个问题是在小平面弯曲之后是否可以跟踪进化。（ii）流体力学中的方程：不可压缩Burgers方程是描述可压缩流体运动的典型粗糙模型。对于Burgers方程，即使初始数据是光滑的，解在有限时间内也会出现跳跃间断。就在这个建议之前，主要研究者引入了一个适当的粘性解的概念来描述一个跳跃的解决方案，这对非发散型方程也很有用。在这个项目中，我们建立了一个奇异的垂直扩散率的概念，这是有用的计算的解决方案在图空间中的数值。特别是我们成功地计算出适当的粘度溶液的水平集方法数值。

Several mathematical foundation is established to investigate the mechanism of variation of shape of solutions due to anisotropy and diffusion effect for nonlinear diffusion equations. We just give two typical topics.(i)Equation describing motion of a crystal surface. When anisotropy of surface energy is strong like snow crystal, there appears a plat surface called a facet in their equilibrium shape. The equations governing its evolution is formally a curvature flow equation. However, the diffusion is so strong that it cannot be viewed as a partial differential equation. We studied a Stefan type free boundary problem describing snow crystal growth assuming that its equilibrium shape is a cylinder. We established (a)Local solvability under the assumption that a facet stays as a facet ; (b)Berg's effect concerning behavior diffusion field near on a facet ; (c)a necessary and sufficient condition for facet bending ; (d)a sufficient condition for existence of a self-similar solution and size condition on facet bending ; (e)a rigorous proof that a facet preserved near equilibrium. A further problem is whether one can track evolution after facet bending.(ii)Equations in fluid mechanics : An invicid Burgers equation is a typical coarse model to describe motion of compressible fluid. The solution develops jump discontinuity in finite time even if the initial data is smooth for the Burgers equation. Just before this proposal, the principal inverstigator introduced a notion of a proper viscosity solution to describe a solution with jumps which is also useful for equations with nondivergence type. In this project we established a notion of singular vertical diffusivity which is useful to calculate the solution numerically in the graph space. In particular we are successful to calculate a proper viscosity solution by a level set method numerically.