Nonlinear Analysis on diffusion effects producing singular shapes

扩散效应在非线性非平衡现象中起着关键作用。为了理解它的作用，我们主要研究了以下主题（i）具有非局部曲率的界面动力学（ii）具有高阶扩散的界面动力学（iii）折叠能量（iv）Navier-Stokes初值问题。(i)在界面控制模型中，晶体表面的演化由带驱动力的曲率流方程描述。有几种情况下，界面能的各向异性是如此之强，其武尔夫形状有一个平坦的部分称为小平面。在这种情况下，甚至解的概念也不清楚，因为扩散效应被认为是非局部的。对于演化曲线，我们通过扩展水平集方法引入了解的概念，并证明了比较定理和收敛定理。作为结果的应用，我们证明了一般方程的晶体算法的收敛性，并证明了晶体运动是光滑界面能运动的极限。由于曲率效应是非局部的，这些工作是高度非平凡的。为了完成这个理论，我们发表了200多页。(ii)通过表面扩散的运动包括四阶扩散效应。与二阶模型相比，其解的行为有很大的不同。例如，我们证明了一个解在有限时间内既可以是松凸的，也可以是非自相交的。(iii)该能量被认为是密度包含二阶导数的能量的极限。我们已经证明了它的较低的连续性，这引发了许多其他工作在这个方向。(iv)我们证明了光滑整体平面Navier-Stokes流的唯一存在性，即使初始数据仅仅是有界的，许多在空间无穷远不衰减。除了这些工作，我们已经出版了一本书“非线性偏微分方程”（共立），包括主题的捏现象的平均流量和大时间行为的解决方案的涡度方程与几个新的结果。

Diffusion effects often play a key role in nonlinear nonequilibrium phenomena. To understand its role we mainly studied the following topics (i) interfacial dynamics with nonlocal curvature (ii) interface dynamics with higher order diffusion (iii) fold energy (iv) the Navier-Stokes initial value problem.(i) Evolution of crystal surface is described by a curvature flow equation with a driving force in the interface controlled models. There are several situations that anisotropy of interfacial energy is so strong that its Wulff shape has a flat portion called a facet. In this case even the notion of solution was unclear since the diffusion effect is considered nonlocal. For an evolving curve we introduce a notion of solutions by extending a level set method and prove the comparison and convergence theorems. As applications of our results we prove the convergence of crystalline algorithm for general equations and that the crystalline motion is a limit of the motion by smooth interfacial energy. These works are highly nontrivial since curvature effect is nonlocal. To complete the theory we published more than 200 pages.(ii) Motion by surface diffusion includes a fourth order diffusion effect. Compared with second order model, the behavior of solution is quite different. For example, we proved that a solution may loose convexity as well as non self-intersection properly in a finite time.(iii) This energy is considered as a limit of energy whose density include the second order derivatives. We have proved its lower semicontinuity which triggers a lot of other works in this direction.(iv) We have proved the unique existence of smooth global planer Navier-Stokes flow even when initial data is merely bounded, many not decay at space infinity. Besides these works we have published a book "Nonlinear Partial Differential Equation" (Kyoritsu) including topics on pinching phenomena of mean flow and large time behavior of solutions of vorticity equations with several new results.