Dynamic Scaling, Coarsening and Stability in Physical Systems

拟开展的工作的主题是理解扩展物理系统中的动态标度现象，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基础上。动力学标度律的矛盾在于，它们通常可以通过包络线计算来预测，但多年来却无法进行严格的分析。我们将研究当前感兴趣的各种问题中的标度和稳定性：Smoluchowski平均场凝聚模型中自相似行为的趋势;粘性流体流动中指状图案的标度和稳定性;相变中畴粗化的模型;孤立波的存在性和稳定性的无限维哈密顿系统和稳定性的玻色-爱因斯坦凝聚与涡。研究内容包括非线性系统稳定性分析的改进方法、自相似解和非线性波的严格分析和数值计算方法、晶体表面台阶结构的物理模拟方法等。 我们的目标是开发新的和有效的数学方法和概念的建模和分析的动态行为的各种问题的高度科学兴趣。我们研究的基本数学模型的大规模动态：小颗粒的团聚;粘性流体流动的混合模式;金属合金和半导体表面的微观结构;流体界面上的波;和超冷气体中的量子涡旋。 将被调查的许多问题是凝聚态物理学，气溶胶物理学，化学工程和材料科学的根本利益。

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.