Free Boundary Problems for Aggregation Phenomena and other Partial Differential Equations

自组织，即一个群体成员之间的局部互动中出现的一种集体行为，在应用科学中普遍存在。例如，一些细菌被化学信号吸引到彼此之间，并可以形成巨大的聚集在一起的集群，充当新的超级有机体。这种聚集的能力对于它们的生存和繁殖能力是必不可少的。对这些细菌行为的数学描述类似于其他自组织现象，如鸟类的聚集行为或拥挤的人群运动，类似的概念已被用于模拟肿瘤生长。在所有情况下，凝聚力群体的形成都是成员之间长期吸引和短期排斥相互作用竞争的结果。研究人员将研究一类数学模型的一对一局部相互作用和由此产生的集体运动之间的关系，这些模型考虑了这两种相互竞争的力量。这些模型通常是复杂的偏微分方程组，描述了单个成员的运动或成员的密度分布函数。这项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几何类型的有效模型。而且，为了使用这些更简单的模型来从理论和数值上研究细菌种群的长期动态，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。研究人员将主要研究可以识别高聚集密度和低聚集密度的界面分隔区域的模型(相分离)。因此，虽然起始点是描述密度函数演化的偏微分方程组，但最终的集体运动是通过描述界面演化的自由边界近似来建模的。本课程将使用偏微分方程组理论、变分、最优运输和几何测度论等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，证明非局部吸引行为在界面上具有与表面张力相同的平滑效果(在适当的比例下)。将首先对宏观模型(例如扩散-聚集方程)进行渐近分析，然后对诸如动力学方程之类的介观模型进行渐近分析。了解如何在动力学模型中解释拥堵效应是本研究的一个重要方面。研究人员还将推导和研究模拟细胞运动的自由边界近似。对这些模型的严格分析将建立不稳定性和对称性破坏性质，这些性质与细胞的行为(所谓的细胞自极化)相对应。最后，这里讨论的许多模型都有一个特殊的结构：它们是相对于沃瑟斯坦距离的梯度流-这是通过最优运输理论定义的。研究人员将探索在离散环境下最优运输的规则性理论的发展。这是在最优交通领域开发有效的数值方法的重要一步，适用于上面讨论的模型。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.