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，并将其应用于科学的各个领域。PI计划解决的许多问题都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。要研究的问题有一定的普遍性，在这个意义上，同样的范式再次出现从几何和分析，流体动力学和材料科学，金融数学，最近，生物学和随机几何。该项目为研究生和博士后研究人员提供研究培训机会。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推移阻塞多孔介质的“历史”时（Caputo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，特别是当它通过两种具有不同孔隙度（传输条件）的介质时。相关的数学现象涉及通过半透膜的盐水流动（其中盐只能在一个方向上通过膜流动）。另一个由全球考虑引起的流动的例子涉及准地转方程：这个方程描述了海洋表面温度的演变，这里温度在一定距离上受到大气的影响。PI与一名研究生合作，正在研究这一过程在陆地-海洋界面的时间演变。考虑的另一个有趣的问题与“重叠”的相互作用有关。一个例子是购买与销售商品的定价结构。有一个区域的“价值”的基础好，这两个价格分歧，和一个他们走到一起。在数学上，这些都是非常有趣的问题有关的稳定性的配置，特别是边缘划分一个行为从其他在一个非线性的方式。在另一个领域，PI正在研究隔离和捕食者-被捕食者模型的问题。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.