Nonlinear Nonlocal Aggregation-Diffusion Partial Differential Equations and Applications

本提案将侧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（PDEs）的广泛领域内的非线性非局部扩散和动力学方程领域的一些选择的新应用中进行数值模拟。在众多应用领域中，我们将特别关注一些例子，这些例子可以在建模阶段被识别为由大量显示“集体行为”的“个体”组成的系统，以及如何从它们中获得“平均”信息。个体的行为通常可以通过随机/确定性偏微分方程来建模，从中人们可以获得基于平均场型偏微分方程的宏观和/或宏观描述，从而得到动力学和/或连续模型系统。聚集/相互作用行为（非局部、非线性）、输运现象和非线性扩散之间的相互作用是本提案分析的主要目标。研究的重点是开发工具，从应用分析的角度来理解与这些方程相关的长时间渐近性、模式的稳定性和函数不等式。另一方面，开发数值方案来准确地解决这些模型将有助于理解这些理论问题，同时为拟议的应用提供信息。该提案是几个数学子领域的焦点，研究课题需要的工具和思想范围从微分几何到使用概率论的数学分析，并通过建模和数值分析。它还涉及当今科学技术的不同核心领域，如数学生物学中的基于代理的模型。重点从偏微分方程的角度进行了应用和数值分析。

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.