Nonlinear PDE models for aggregation phenomena

提出的研究重点是对聚集现象的数学模型进行理论和数值研究。这种现象出现在群体生物学（细胞趋化性，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。本文将特别关注两类非线性和非局部偏微分方程模型。第一个是宏观密度的主动输运方程，其中速度与它有函数依赖，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，其中左侧表示平流，右侧包含增益和损失项（由于转弯）。后一种模型认为转动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)社会互动和非线性/分数扩散之间的微妙平衡在创建上下文相关的解决方案中，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。我们感兴趣的是解决方案的完备性和它们的长时间行为，以及模型捕获的模式形成和自集体行为。还将考虑聚集模型的新应用（如蛋白质在表面上的吸附），并将寻求符合最近对鱼群的实验的新模型。

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.