Free Boundary Problems for Cell Motility and Other Applications

这个项目致力于研究一些在物理和生物学中有重要应用的数学问题。其中一个项目是对真核细胞在底物上的运动进行建模。细胞运动参与伤口愈合和免疫反应等关键生理过程。真核细胞最显著的特征之一是它们能够以一种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持续运动(细胞迁移)。虽然涉及的生物过程非常复杂，但首席研究员将开发和研究一些数学模型(通过生物学家提出的模型的近似获得)，这些模型更易于分析，数值计算也更快。通过识别导致细胞迁移的模型，这项工作将有助于更好地理解哪些生物过程在细胞运动中发挥关键作用。另一个不同的项目是研究最优运输问题解的精细性质。最优运输问题是一类数学问题，它起源于一个简单的问题，即如何以最便宜(或最有效)的方式将一组来源的生产最佳分配到一组目的地。这一数学领域在经济学、数据分析、图像处理等多个领域都有应用。除了理论研究外，本研究还将有助于这些复杂问题的解的数值计算。通过积极参与该项目，对研究生进行培训。上面描述的第一个项目涉及Hele-Shaw型自由边界问题，其中通常的平均曲率的平滑/稳定效应被有源势的不稳定效应所平衡。研究者将研究对称破缺分支现象，其特征是此类问题的非平凡行波解的存在性。此外，还将研究拥挤人群运动建模和流体动力学中出现的相关自由边界问题。提案的重点是移动边界的向前和向后运动通过不同的机制和不同的时间尺度发生的模型/制度。在最优公共交通领域，焦点是与绝对连续措施的离散近似的措施相关的最优计划的性质。这样的框架在许多应用中是非常重要的，特别是对于数值计算。我们将发展相关的Kantorovich势的正则性理论。最后一个项目涉及非局部方程(例如分数阶拉普拉斯方程)的边界条件，它们比局部方程更精细是出了名的。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。该奖项反映了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。

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.