Non-Linear Diffusion Modeling: From Geometry, to Materials, to Social Dynamics

首席研究员（PI）将研究一系列数学问题，并将其应用于不同的科学领域。PI计划解决的许多问题是他最近探索的问题的自然延续，这些问题在更广泛的数学社区中不断发展和扩散。要研究的问题具有一定的普遍性，因为从几何和分析、流体动力学和材料科学、金融数学以及最近的生物学和随机几何中都出现了相同的范式。本项目为研究生和博士后提供科研培训机会。第一个项目涉及多孔介质中的可压缩流动（例如，气体），特别是当流动具有随时间阻塞多孔介质的“历史”时（卡普托扩散模型）。PI和一名高级研究生正在研究Caputo扩散模型的一些重要方面，特别是当它发生在两种不同孔隙率（传输条件）的介质中。相关的数学现象涉及到盐通过半透膜的流动（在这种情况下盐只能沿一个方向流过膜）。另一种由全局考虑引起的流动与准地转方程有关：该方程描述了海洋表面温度的演变，在这里温度通过大气受到一定距离的影响。在与一名研究生的合作下，PI正在研究这一过程在陆海间期的时间演变。另一个有趣的问题与“重叠”交互有关。一个例子是商品买卖的定价结构。在一个潜在商品的“价值”区域，这两种价格是不同的，而在另一个区域，它们是一致的。从数学上讲，这些都是非常有趣的问题，与构型的稳定性有关，特别是对于以非线性方式将一种行为与另一种行为分开的边缘。在另一个领域，PI正在研究隔离问题和捕食者-猎物模型。该奖项反映了美国国家科学基金会的法定使命，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。

The principal investigator (PI) will investigate a series of problems in mathematics with applications to diverse areas of Science. A good number of the issues the PI plans to address are a natural continuation of problems he has explored recently that keep progressing and diffusing through a wider mathematical community. The questions to be studied have a certain universality in the sense that the same paradigm reappears from geometry and analysis, to fluid dynamics and material sciences, to financial mathematics and, more recently, biology and stochastic geometry. The project provides research training opportunities for graduate students and postdoctoral researchers.The first project concerns compressible flow (for instance, a gas) in porous media, in particular when the flow has a "history" that clogs with time the porous media (the Caputo diffusion model). Some important aspects of the Caputo diffusion model are being studied by the PI and an advanced graduate student, in particular when it occurs through two media with different porosity (transmission condition). Related mathematical phenomena concern the saline flow through semipermeable membranes (where the salt can flow only in one direction through the membrane). Another case where the flow is induced by global considerations concerns the quasi-geostrophic equation: this equation describes the evolution of the temperature on the surface of the ocean, and here temperature is influenced at a distance though the atmosphere. In collaboration with a graduate student the PI is working on the time evolution of this process in the land-sea interphase. Another interesting problem considered is concerned with "overlapping" interactions. An example is the structure of pricing for buying vs selling of goods. There is an area of "values" of the underlying good where the two prices diverge, and one where they come together. Mathematically, these are very interesting problems concerned with the stability of the configuration, in particular for the edge dividing one behavior from the other in a nonlinear way. In a different area, the PI is studying issues of segregation and predator-prey models.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.