Scaling dynamics and stability in extended physical systems

PegoDMS-0905723 这项工作的重点是理解复杂非线性系统的稳定动力学行为。 研究了具有强烈物理意义的模型中的各种现象，特别是：聚集和粗化模型中的动力学标度（含输入的凝聚方程，冶金学中的简化晶粒生长模型，含软碰撞的弹道聚合）;非线性场论中相干结构的鲁棒性（特别是精确孤立水波的谱稳定性）;和不可压缩粘性流体的稳定性分析，根据压力边界条件的最新进展（解决域与角落，耦合到磁性和应力松弛，和零粘度极限）。动态标度的工作特别旨在进一步发展将动力系统方法与概率中的无限可分性理论联系起来的思想。 该计划的进展导致改进的分析方法，以了解相干行为如何出现在复杂的非线性系统。 数学上研究的特殊模型具有基本的科学意义，与纳米材料科学、气溶胶物理学（烟雾、煤烟和烟雾的模型）、种群遗传学、物理化学和其他领域有关。 对流体边界的研究应该会导致小尺度流动的数值模拟的改进，从而对研究小型飞行和游泳物体、血液流动和工程微流体系统产生影响。

PegoDMS-0905723 The focus of this work is on understanding stable dynamicalbehavior in complex nonlinear systems. A variety of phenomenaare investigated in models of strong physical interest,especially: dynamic scaling in models of aggregation andcoarsening (coagulation equations with input, simplified graingrowth models in metallurgy, ballistic aggregation with softcollisions); the robustness of coherent structures in nonlinearfield theories (in particular, spectral stability for exactsolitary water waves); and stability analysis for incompressibleviscous fluids in light of recent progress on boundary conditionsfor pressure (addressing domains with corners, coupling tomagnetism and stress relaxation, and the zero-viscosity limit). Work on dynamic scaling in particular aims to further developideas that connect dynamical systems methods with infinitedivisibility theory in probability. Progress in this program leads to improved mathematicalmethods for understanding how coherent behavior emerges incomplex nonlinear systems. The particular models to be studiedmathematically are of fundamental scientific interest, relevantto materials science at the nanoscale, aerosol physics (models ofsmog, soot, and smoke), population genetics, physical chemistry,and other fields. The work on fluid boundaries should lead toimprovements in the numerical modeling of small-scale flows, withconsequences for studying small flying and swimming objects,blood flow, and engineered microfluidic systems, for example.