Dynamic Scaling, Coarsening and Stability in Physical Systems

这项工作的主要主题是理解扩展物理系统中普遍存在的动态标度现象，建立在自相似解、非线性波脉冲和涡旋等相干结构的稳定性分析的基础上。动态标度律的悖论是，它们通常可以通过包络计算的支持来预测，但却无法进行多年的严格分析。我们将研究各种令人感兴趣的问题的标度和稳定性：Smoluchowski的平均场凝聚模型中的自相似行为的趋势；粘性流体流动中指纹图的标度和稳定性；相移中的区域粗化模型；无限维哈密顿系统中孤立波的存在性和稳定性；以及带涡旋的玻色-爱因斯坦凝聚体的稳定性。这项工作将包括发展改进的方法，用于非线性系统的稳定性分析，严格的自相似解和非线性波的分析和数值计算，以及晶体表面的阶梯结构的物理模拟。这项工作的总体目的是提高对复杂系统中大尺度动力学行为的理解。我们的目标是开发新的有效的数学方法和概念，用于对各种具有高度科学兴趣的问题的动态行为进行建模和分析。我们研究了大尺度动力学的基本数学模型：小颗粒的团聚；粘性流体流动中的混合模式；金属合金和半导体表面的微观结构；流体界面上的波；以及超冷气体中的量子涡旋。许多将被研究的问题是凝聚态物理、气溶胶物理、化学工程和材料科学的基本兴趣。

The main theme of the proposed work is on understanding the widespreadphenomenon of dynamic scaling in extended physical systems, buildingon stability analysis of coherent structures such as self-similarsolutions, nonlinear wave pulses and vortices. The paradox of dynamicscaling laws is that typically they can be predicted by back of theenvelope calculations, yet defy rigorous analysis for years.We will study scaling and stability in a variety of problems of highcurrent interest: the trend to self-similar behavior in Smoluchowski'smean-field model of coagulation; scaling and stability of fingeringpatterns in viscous fluid flows; models of domain coarsening in phasetransitions; existence and stability of solitary waves ininfinite-dimensional Hamiltonian systems; and stability of Bose-Einstein condensates with vortices. The work will involve the development of improved methods for stability analysis in nonlinear systems, for the rigorous analysis and numerical computation of self-similar solutions and nonlinear waves, and for physical modeling of step-terrace structures on crystalline surfaces.The general aim of this work is to improve understanding of large-scaledynamic behavior in complex systems. We aim to develop new and effective mathematical methods and concepts for the modeling and analysis of dynamic behavior in a variety of problems of high scientific interest. We study fundamental mathematical models for the large-scale dynamics of: agglomeration of small particles; mixing patterns in viscous fluid flow; microscopic structure of metallic alloys and semiconductor surfaces; waves on fluid interfaces; and quantum vortices in super-cold gases. Many issues that will be investigated are of fundamental interest in condensed-matter physics, aerosol physics, chemical engineering, and materials science.