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.

该项目致力于研究在物理和生物学中具有重要应用的一些数学问题。其中一个项目与真核细胞在底物上运动的建模有关。细胞运动与关键生理过程有关，例如伤口愈合和免疫反应。真核细胞最引人注目的特征之一是它们以看似自发的方式达到和保持不对称形状的能力，这种现象导致细胞沿给定方向持续运动（细胞迁移）。尽管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），这些模型既易于分析，又可以更快地进行计算。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起着关键作用。一个不同的项目旨在研究最佳运输问题解决方案的良好特性。最佳运输问题是一类数学问题，其起源于如何以最便宜（或最有效）方式将生产从一组来源分配到一组目的地的简单问题。该数学领域在各种领域（例如经济学，数据分析，图像处理等）都有应用。除了理论研究外，此研究还将有助于这些复杂问题解决方案的数值计算。研究生将通过积极参与该项目进行培训。上面描述的第一个项目涉及Hele-Shaw类型的自由边界问题，在这种问题中，均值曲面的通常平滑/稳定效应通过主动潜力的不稳定效应来平衡。研究者将研究对称性破坏分叉现象的特征，其特征是存在于此类问题的非平凡循环解决方案。还将研究相关的自由边界问题，这些问题在拥挤的人群运动和流体动态的建模中也会受到研究。该提案的重点是模型/制度，其中移动边界的前向和向后运动通过不同的机制和不同的时间尺度发生。在最佳质量运输领域，重点是与绝对连续度量的离散近似措施相关的最佳计划的特性。这样的框架在许多应用中非常重要，尤其是数值计算。将开发针对Kantorovich潜力的规则性理论。最终项目与非局部方程的边界条件有关（例如，分数拉普拉斯方程），众所周知，它们比本地对应物更精致。该项目的主要目的是推导新的非本地诺伊曼边界条件，这是气体动力学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的法定任务，并被认为是通过使用评估的支持值得的。基金会的智力优点和更广泛的影响评论标准。