Dynamic Scaling, Coarsening and Stability in Physical Systems

提出的工作的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、非线性波脉冲和漩涡等相干结构的稳定性分析基础上。动态标度定律的矛盾之处在于，它们通常可以通过粗略的计算来预测，但却无法经过多年的严格分析。我们将研究各种高电流问题的标度和稳定性：凝固的Smoluchowski平均场模型中自相似行为的趋势；粘性流体流动中指型的尺度与稳定性相变中畴粗化模型的研究孤立波无穷维哈密顿系统的存在性与稳定性具有涡旋的玻色-爱因斯坦凝聚体的稳定性。这项工作将涉及发展改进的方法，用于非线性系统的稳定性分析，用于自相似解和非线性波的严格分析和数值计算，以及用于晶体表面上台阶-阶地结构的物理建模。这项工作的总体目标是提高对复杂系统中大规模动力学行为的理解。我们的目标是开发新的和有效的数学方法和概念，以模拟和分析各种高科学兴趣问题的动态行为。我们研究了小颗粒团聚大尺度动力学的基本数学模型；粘性流体流动中的混合模式；金属合金和半导体表面的显微组织；流体界面上的波；以及超冷气体中的量子漩涡。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.