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.

自组织是在应用科学中无处不在。例如，某些细菌被化学信号彼此吸引，并且可以形成大型的内聚糖，以充当新的超生物。这种汇总的能力对于它们生存和繁殖的能力至关重要。这些细菌行为的数学描述类似于其他自组织现象的数学描述，例如鸟类的植入行为或拥挤的人群运动，以及类似的思想已被用来模拟肿瘤的生长。在所有情况下，凝聚力组的形成都是成员之间远程有吸引力和短期排斥相互作用之间竞争的结果。研究人员将研究一对一的本地互动与一类数学模型的集体运动之间的关系，这些模型考虑了这两个竞争力量。这些模型通常是偏微分方程的复杂系统，它描述了单个成员或成员密度分布函数的运动。这项研究的目的是通过渐近分析和奇异限制得出描述集体运动的几何类型的新有效模型。并且，要使用这些更简单的模型在理论上和数字上研究细菌种群的长时间动态，预测人群的行为，或者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。研究者将主要研究模型，该模型可以鉴定出高聚集密度和低聚集密度区域的界面（相位分离）。因此，虽然起点是描述密度函数演变的部分微分方程的系统，但所得的集体运动是由描述界面演化的自由边界近似建模的。将使用来自部分微分方程理论，变异，最佳运输和几何测量理论的工具进行严格的数学分析。一个关键目标是提供严格的理由，即非本地吸引力的行为对界面具有与表面张力相同的平滑作用（以适当的比例）。渐近分析将首先在宏观模型（例如扩散 - 聚集方程）上，然后在介观模型（例如动力学方程）上进行。了解在动力学模型中如何解释拥塞效应是这项研究的重要方面。研究者还将得出并研究自由边界近似模拟细胞运动。对这些模型的严格分析将建立不稳定性和对称性破坏特性，这对应于细胞的良好行为（细胞的所谓自动极化）。最后，这里讨论的许多模型都有一个特定的结构：它们是瓦斯汀距离的梯度流 - 这是通过最佳运输理论定义的。研究人员将在离散环境中追求规律性理论以进行最佳运输。这是在最佳运输领域开发有效的数值方法的重要步骤，并适用于上述模型。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来支持的。 。