Nonlinear PDE models for aggregation phenomena

聚合现象的非线性 PDE 模型

基本信息

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

项目摘要

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.

拟议研究的重点是聚集现象数学模型的理论和数值研究。这种现象出现在群体生物学（细胞的趋化性、动物的集群或成群）以及物理和化学（纳米颗粒的自组装、恒星形成等）中。两类非线性和非局部偏微分方程模型将受到特别关注。第一个是宏观密度的主动输运方程，其中速度对其具有函数依赖性，由具有相互作用势的卷积给出。第二个是分布函数的动力学方程，其中左侧代表平流，右侧包含增益和损失项（由于转动）。后一种模型将转向视为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互动和非线性/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）多重物种动力学，v) 扩展到更高维度。我们对解决方案的适定性及其长期行为，以及模型捕获的模式形成和自我集体行为感兴趣。还将考虑聚集模型的新应用（例如表面上的蛋白质吸附），并将寻求符合最近鱼群实验的新模型。