Free Boundary Problems for Aggregation Phenomena and other Partial Differential Equations

聚集现象和其他偏微分方程的自由边界问题

• 批准号: 2307342
• 负责人: Antoine Mellet
• 金额: $ 30万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307342&HistoricalAwards=false
• 关键词: Free; Boundary; Problems; Aggregation; Phenomena

Self-organization, that is the emergence of a collective behavior out of the local interactions between members of a group, is ubiquitous in applied sciences. Some bacteria, for example, are attracted toward each other by chemical signals and can form large cohesive clusters that act as a new super-organism. This ability to aggregate is essential to their ability to survive and proliferate. The mathematical description of these bacteria's behavior is similar to that of other self-organizing phenomena, such as the flocking behavior of birds or congested crowd motion, and analogous ideas have been used to model tumor growth. In all cases, cohesive group formation is the result of the competition between long-range attractive and short-range repulsive interactions between the members. The investigator will study the relationship between the one-to-one local interactions and the resulting collective motion for a class of mathematical models that take in consideration these two competing forces. These models are often complex systems of partial differential equations, which describe the motion of individual members or of the members' density distribution function. The goal of this research is to derive, via asymptotic analysis and singular limits, new effective models of geometric type describing the collective motion. And, to use these simpler models to theoretically and numerically study the long time dynamic of a population of bacteria, predict the behavior of a crowd, or compare the effects of different therapies on tumor growth. The project will offer research training opportunity for students. The investigator will primarily study models for which an interface separating regions of high and low aggregation density can be identified (phase separation). So, while the starting point is a system of partial differential equations that describes the evolution of a density function, the resulting collective motion is modeled by a free boundary approximation describing the evolution of an interface. A rigorous mathematical analysis will be developed using tools from the theory of partial differential equations, the calculus of variation, optimal transportation, and geometric measure theory. A key goal is to provide rigorous justification of the fact that nonlocal attractive behavior has the same smoothing effect on the interface as surface tension (at an appropriate scale). An asymptotic analysis will be performed first on macroscopic models (e.g., diffusion-aggregation equations) and then on mesoscopic models, such as kinetic equations. Understanding how congestion effects can be account for in kinetic models is an important aspect of this research. The investigator will also derive and study free boundary approximations modeling cell motility. The rigorous analysis of these models will establish the instability and symmetry breaking properties, which correspond to well documented behaviors of cells (the so-called self-polarization of cells). Finally, many of the models discussed here have a particular structure: They are gradient flows with respect to the Wasserstein distance - which is defined via the theory of optimal transportation. The investigator will pursue the development of a regularity theory for optimal transportation in a discrete setting. This is an important step toward developing effective numerical methods in the field of optimal transportation, with application to the models discussed above.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

自组织是群体成员在局部相互作用中产生的集体行为，在应用科学中普遍存在。例如，一些细菌通过化学信号相互吸引，可以形成大的凝聚力集群，作为一个新的超级有机体。这种聚集的能力对它们的生存和繁殖能力至关重要。这些细菌行为的数学描述与其他自组织现象类似，例如鸟类的群集行为或拥挤的人群运动，类似的想法已被用于模拟肿瘤生长。在所有情况下，内聚群的形成都是成员之间长程吸引力和短程排斥力相互作用之间竞争的结果。研究人员将研究一对一的局部相互作用和由此产生的集体运动之间的关系，考虑到这两个竞争力的一类数学模型。这些模型通常是复杂的偏微分方程系统，描述单个成员或成员密度分布函数的运动。本研究的目的是通过渐近分析和奇异极限，推导出描述集体运动的新的有效几何模型。并且，使用这些更简单的模型从理论上和数值上研究细菌种群的长期动态，预测人群的行为，或比较不同疗法对肿瘤生长的影响。该项目将为学生提供研究培训机会。研究者将主要研究这样的模型，即可以识别出高聚集密度和低聚集密度的界面分离区域（相分离）。因此，虽然起点是描述密度函数演化的偏微分方程系统，但由此产生的集体运动由描述界面演化的自由边界近似建模。一个严格的数学分析将开发使用工具从偏微分方程理论，变分法，最佳运输，几何测量理论。一个关键的目标是提供严格的理由的事实，即非局部吸引力的行为具有相同的平滑效果的界面上的表面张力（在适当的规模）。首先对宏观模型进行渐近分析（例如，扩散-聚集方程），然后是介观模型，如动力学方程。了解如何拥挤的影响，可以考虑在动力学模型是这项研究的一个重要方面。研究者还将推导和研究自由边界近似模型细胞运动。对这些模型的严格分析将建立不稳定性和对称性破缺性质，这对应于细胞的有据可查的行为（所谓的细胞自极化）。最后，这里讨论的许多模型都有一个特殊的结构：它们是相对于瓦瑟斯坦距离的梯度流-这是通过最优运输理论定义的。研究者将致力于在离散环境中发展最佳运输的规律性理论。这是在最佳运输领域发展有效数值方法的重要一步，并应用于上述模型。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。