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Free Boundary Problems for Cell Motility and Other Applications

细胞运动和其他应用的自由边界问题

基本信息

批准号：
2009236
负责人：
Antoine Mellet
金额：
$ 32.6万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009236&HistoricalAwards=false
关键词：
Free Boundary Problems Cell Motility

项目摘要

This project is devoted to the study of some mathematical problems that have important applications in physics and biology. One of the projects is concerned with the modeling of the motion of eukaryotic cells on a substrate. Cell motility is involved in key physiological processes such as wound healing and immunological response. One of the most remarkable characteristics of eukaryotic cells is their ability to reach and maintain an asymmetric shape in a seemingly spontaneous way, a phenomenon that leads to the sustained motion of the cell in a given direction (cell migration). While the biological processes involved are very complex, the principal investigator will develop and study some mathematical models (obtained as approximation of models proposed by biologists) that are both easier to analyze and faster to compute numerically. By identifying models that lead to cell migration, this work will help better understand what biological processes play a key role in cell motility. A different project is aimed at the study the fine properties of the solutions of optimal transportation problems. Optimal transportation problems are a class of mathematical problems that originated with the simple question of how to optimally allocate the production from a set of sources to a set of destinations in the cheapest (or most efficient) way. This field of mathematics has application in a variety of domains such as economics, data analysis, image processing etc. In addition to theoretical studies this research will contribute to the numerical computations of the solutions of these complex problems. Graduate students will be trained through active participation in the project. The first project described above involves free-boundary problems of Hele-Shaw type in which the usual smoothing/stabilizing effect of mean-curvature is balanced by the destabilizing effect of an active potential. The investigator will study symmetry breaking bifurcation phenomena characterized by the existence of nontrivial traveling wave solutions for such problems. Related free-boundary problems, arising in the modeling of congested crowd motion and in fluid dynamic will also be studied. The focus of the proposal is on models/regimes in which the forward and backward motions of the moving boundary occur via different mechanisms and at different time scales. In the field of optimal mass transportation, the focus is on the properties of optimal plans associated to measures that are discrete approximation of absolutely continuous measures. Such a framework is of great importance in many applications and in particular for numerical computations. A regularity theory for the associated Kantorovich potential will be developed. The final project is concerned with boundary conditions for nonlocal equations (e.g. fractional Laplace equation), which are notoriously more delicate than their local counterparts. The main goal of this project is derivation of new nonlocal Neumann boundary conditions, which are the macroscopic counterparts of classical microscopic boundary conditions in the kinetic theory of gas dynamic.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.

这个项目致力于研究一些在物理学和生物学中有重要应用的数学问题。其中一个项目是关于真核细胞在基底上运动的建模。细胞运动参与关键的生理过程，如伤口愈合和免疫反应。真核细胞最显著的特征之一是它们能够以看似自发的方式达到并保持不对称的形状，这种现象导致细胞在给定方向上的持续运动（细胞迁移）。虽然所涉及的生物过程非常复杂，但主要研究者将开发和研究一些数学模型（作为生物学家提出的模型的近似值），这些模型更容易分析，计算速度更快。通过识别导致细胞迁移的模型，这项工作将有助于更好地了解哪些生物过程在细胞运动中发挥关键作用。另一个不同的项目旨在研究最优运输问题解的优良性质。最优运输问题是一类数学问题，起源于如何以最便宜（或最有效）的方式将生产从一组来源最优分配到一组目的地的简单问题。这一领域的数学应用在各种领域，如经济学，数据分析，图像处理等，除了理论研究，这项研究将有助于这些复杂问题的解决方案的数值计算。研究生将通过积极参与该项目接受培训。上述第一个项目涉及Hele-Shaw型的自由边界问题，其中平均曲率的通常平滑/稳定效应被活动势的不稳定效应所平衡。研究人员将研究对称性破缺分岔现象，其特征在于存在非平凡行波解。相关的自由边界问题，在拥挤的人群运动的建模和流体动力学中产生的也将进行研究。该提案的重点是通过不同的机制和在不同的时间尺度上发生移动边界的向前和向后运动的模型/制度。在最优质量运输领域，重点是与绝对连续度量的离散近似度量相关联的最优计划的属性。这样一个框架是非常重要的，在许多应用中，特别是数值计算。一个规律性的理论相关联的康托洛维奇潜力将被开发。期末专题是关于非局部方程（例如分数阶拉普拉斯方程）的边界条件，这是出了名的比它们的局部方程更精细。该项目的主要目标是推导新的非局部Neumann边界条件，这是气体动力学理论中经典微观边界条件的宏观对应物。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。