Mathematical Analysis on Change of Patterns by Anisotropy and Diffusion

建立了几个数学基础来研究各向异性和扩散效应引起的非线性扩散方程解的形状变化机制。我们只给出两个典型的主题：(I)描述晶体表面运动的方程。当表面能的各向异性很强时，如雪晶，就会出现一个平面，称为小面，它的平衡形状。控制其演化的方程形式上是曲率流动方程。然而，扩散是如此强烈，以至于它不能被视为偏微分方程式。我们研究了一个描述雪晶生长的Stefan自由边界问题，假定它的平衡形状是一个圆柱体。我们建立了(A)小平面作为小平面的假设下的局部可解性；(B)关于小平面附近行为扩散场的Berg效应；(C)小平面弯曲的充要条件；(D)小平面弯曲存在自相似解的充分条件；(E)小平面保持在平衡附近的严格证明。(Ii)流体力学方程：一个不变的Burgers方程是描述可压缩流体运动的一个典型的粗模型。即使Burgers方程的初始数据是光滑的，解也会在有限时间内产生跳跃间断。就在这个建议之前，主逆变者引入了一个适当粘性解的概念来描述有跳跃的解，这对非散度型方程也很有用。在这个项目中，我们建立了奇异垂直扩散系数的概念，这对于在图空间中数值计算解是有用的。特别是，我们成功地用Level Set方法数值计算了适当的粘性解。

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.