Nonlinear PDE models for aggregation phenomena

聚合现象的非线性 PDE 模型

• 批准号: 341834-2013
• 负责人: Fetecau, Razvan
• 金额: $ 0.8万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572376
• 关键词: Nonlinear; PDE; models; aggregation; phenomena

The focus of the proposed research is a theoretical and numerical investigation of mathematical models for aggregation phenomena. Such phenomena arise in population biology (chemotaxis of cells, swarming or flocking of animals), as well as in physics and chemistry (self-assembly of nanoparticles, stellar formation, etc). Two classes of nonlinear and nonlocal PDE models will be given particular attention. The first is an active transport equation for the macroscopic density, where the velocity has a functional dependence on it, given by a convolution with an interaction potential. The second is a kinetic equation for the distribution function, where the left-hand-side represents advection and the right-hand-side contains gain and loss terms (due to turning). The latter model regards turning as the fundamental mechanism for animal aggregation. We plan to investigate several aspects related to these two models: i) the delicate balance between social interactions and nonlinear/ fractional diffusion in creating context-relevant solutions, ii) the role of boundaries, iii) the effect of anisotropy, iv) multiple-species dynamics, v) extensions to higher dimensions. We are interested in the well-posedness of solutions and their long time behaviour, as well as the pattern formation and self-collective behaviours captured by the models. New applications of aggregation models (such as protein adsorption on a surface) will also be considered, and new models that conform to recent experiments on fish schools, will be sought.

建议的研究重点是聚集现象的数学模型的理论和数值研究。这种现象出现在种群生物学（细胞的趋化性，动物的群集或群集）以及物理和化学（纳米粒子的自组装，恒星形成等）中。 两类非线性和非局部偏微分方程模型将给予特别关注。第一个是宏观密度的主动输运方程，其中速度对它有函数依赖性，由与相互作用势的卷积给出。 第二个是分布函数的动力学方程，其中左手边表示平流，右手边包含增益和损失项（由于转向）。后一种模型认为转向是动物聚集的基本机制。 我们计划研究与这两个模型相关的几个方面：i）在创建上下文相关解决方案时，社会相互作用和非线性/分数扩散之间的微妙平衡，ii）边界的作用，iii）各向异性的影响，iv）多物种动力学，v）向更高维度的扩展。我们感兴趣的解决方案的适定性和他们的长期行为，以及模式的形成和自集体行为捕获的模型。 还将考虑聚集模型的新应用（如蛋白质吸附在表面上），并将寻求符合最近鱼群实验的新模型。