Nonlinear Nonlocal Aggregation-Diffusion Partial Differential Equations and Applications

本计划将着重于发展新的数学分析工具和方法，设计合适的数值格式，并在偏微分方程（PDE）的广泛领域内对非线性非局部扩散和动力学方程的一些选定的新应用进行数值模拟。在众多的应用领域中，我们将特别集中在一些例子上，这些例子在建模阶段可以被识别为由大量的“个体”组成的系统，这些个体表现出“集体行为”，以及如何从这些个体中获得“平均”信息。个人的行为可以典型地通过随机/确定性ODE建模，从中获得基于平均场型PDE的宏观和/或宏观描述，从而导致动力学和/或连续模型系统。之间的相互作用的聚集/相互作用的行为（非本地的，非线性的），运输现象，和非线性扩散，是本proposal.The研究的分析的主要目标是开发工具，以了解长期的渐近性，稳定性的模式，和功能的不平等关系到这些方程从thesapplied分析的观点。另一方面，发展精确求解这些模型的数值方法将有助于理解这些理论问题，同时为拟议的应用提供信息。这一建议是几个数学子领域的焦点，研究课题需要从微分几何到数学分析的工具和思想，使用概率论，并通过建模和数值分析。它也触及不同的核心领域，如今的兴趣在科学和技术，如代理为基础的模型在数学生物学.重点是从偏微分方程的角度进行应用和数值分析.

This proposal will focus on the development of new mathematicalanalysis tools and methods, design of suitable numerical schemes,and numerical simulation in some selected new applications of thefield of nonlinear nonlocal diffusion and kinetic equations insidethe broad area of Partial Differential Equations (PDEs). Among thenumerous areas of applications, we will concentrate particularlyon some examples which can be identified, at the modelling stage,as systems made out of a large number of "individuals" which showa "collective behaviour" and how to obtain from them "averaged"information. The behaviour of individuals can be typicallymodelled via stochastic/deterministic ODEs from which one obtainsmesoscopic and/or macroscopic descriptions based on mean-fieldtype PDEs leading to kinetic and/or continuum model systems. Theinterplay between the aggregation/interaction behaviour (nonlocal,nonlinear), the transport phenomena, and the nonlinear diffusion,is the main goal of analysis of this proposal.The research to be developed is centered on developing tools tounderstand the long time asymptotics, stability of patterns, andfunctional inequalities related to these equations from theapplied analysis viewpoint. On the other hand, developingnumerical schemes to solve accurately these models will helpunderstanding these theoretical issues while giving informationfor the proposed applications. This proposal is a focal point ofseveral mathematical subareas, the research topics need tools andideas ranging from differential geometry to mathematical analysisusing probability theory and passing through modeling andnumerical analysis. It also touches different core areas ofnowadays interest in Science and Technology such as agent-basedmodels in mathematical biology. The emphasis of the proposal is inapplied and numerical analysis from a Partial DifferentialEquations perspective.