Free Boundary Problems and Other Partial Differential Equations

这个研究项目的一个方面涉及所谓的自组织现象的数学分析，即一个大群的代理（可能是鱼，鸟，分子等）行为作为一个连贯的结构，没有一个领导者的方向，但通过只有本地的相互作用，个别代理与他们最近的邻居。该项目的目标是调查代理之间的一对一的互动如何决定群体的全局行为。该项目的第二个方面涉及固体载体上小液滴运动的数学研究。已经开发了各种模型来描述这种现象;该项目研究了其中几种模型的详细数学特性，作为更好地理解哪些模型可以用于准确描述液滴的特定行为的一步。特别令人感兴趣的是接触线（空气、液体和固体相遇的界面）的几何性质。最后，该项目还研究了与通过某些材料（主要是绝缘晶体）的热传播建模相关的数学问题。挑战之一是更好地理解描述晶体原子运动的微观定律如何产生热扩散的经典宏观模型（如经典傅立叶定律）。该项目研究的大多数数学问题都是自由边界问题（在先验未知但必须作为解的一部分找到的域上求解的偏微分方程）。这是特别是移动液滴问题的情况，其中自由边界是上面提到的接触线。该项目研究了各种模型（对应于自由边界条件的不同选择）如何影响接触线的规则性。该项目还研究了简单的吸引/排斥机制下的自组织现象与一些阶数高于2的障碍问题（如双调和障碍问题）的精细正则性之间的关系。调和障碍问题是一个经典的自由边界问题，但其高阶问题仍有许多未解之谜。许多经典工具（如最大值原理）似乎不再相关，而其他工具，特别是几何测度理论的工具，应该发挥重要作用。该项目还研究了模拟相分离现象的相关演化问题（导致空区域分离的正浓度补丁）。所研究的方程让人想起Cahn-Hilliard方程，但涉及退化和非局部扩散现象。最后，该项目探索了从热传播（和其他传输现象）动力学模型中严格推导非局部积分-微分方程。一个这样的动力学模型是玻尔兹曼声子方程。与致力于稀气体玻尔兹曼方程的大量工作相比，对该方程的研究仍处于早期阶段，该项目将有助于该理论的发展。

One aspect of this research project concerns the mathematical analysis of so-called self-organizing phenomena, that is, the process by which a large group of agents (which could be fish, birds, molecules, etc.) behaves as a coherent structure without the direction of a leader but through only local interactions of individual agents with their nearest neighbors. The goal of the project is to investigate how the one-to-one interactions between the agents determine the global behavior of the group. A second aspect of this project concerns the mathematical study of the motion of small droplets of liquid on a solid support. Various models have been developed to describe this phenomenon; this project investigates the detailed mathematical properties of several of these models, as a step toward better understanding which models can be used to accurately describe specific behaviors of the droplets. Of particular interest are the geometric properties of the contact line (the interface where air, liquid and solid meet). Finally, this project also investigates mathematical problems related to the modeling of heat propagation through certain materials (mostly insulating crystals). One of the challenges is to better understand how the microscopic laws, which describe the motion of the atoms of the crystals, yield the classical macroscopic models for heat diffusion (such as the classical Fourier law). Most of the mathematical problems under study in this project are free boundary problems (partial differential equations to be solved on domains that are not known a priori, but must be found as part of the solution). This is the case in particular of the moving drop problem, where the free boundary is the contact line mentioned above. The project investigates how various models (corresponding to different choices of free boundary conditions) affect the regularity of the contact line. The project also studies the relationship of self-organizing phenomena under simple attraction/repulsion mechanisms to fine regularity properties for some obstacle problems of order higher than two (such as the biharmonic obstacle problem). While the harmonic obstacle problem is a classical free boundary problem, there are still many unanswered questions concerning its higher order counterpart. Many classical tools (such as the maximum principle) seem no longer relevant, while other tools, in particular tools from geometric measure theory, should play an important role. The project also investigates related evolution problems that model the phenomenon of phase separation (leading to patches with positive concentration separated by empty regions). The equations under study are reminiscent of Cahn-Hilliard equations, but involve degenerate and non-local diffusion phenomena. Finally, the project explores the rigorous derivation of non-local integro-differential equations from kinetic models for heat propagation (and other transport phenomena). One such kinetic model is the Boltzmann phonon equation. The study of this equation is still in its early stage compared to the considerable amount of work devoted to the Boltzmann equation for dilute gases, and the project will contribute to the development of this theory.