Nonlinear PDE models for aggregation phenomena

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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.