Dynamic Free-Boundary Problems

“动态自由边界问题”是在演化未知的域中寻找偏微分方程的解的问题。一个例子是冰融化建模问题，其中冰和水的界面是由水区域的温度分布（即热方程的解）动态确定的。该项目解决了有关自由边界问题的一些基本问题，例如解的存在性和长期行为。这是一个重要的话题，因为如果没有适当的数学理论，就很难开发出准确且值得信赖的数值方法来处理具体的物理问题。一个很好的例子是液滴在倾斜表面上滑动的问题。在这种情况下，当速度增加时，液滴的形状会发生巨大变化，并且当速度超过某个临界值时，液滴的后部会形成奇点（角点）。在较高的速度下，液滴的尾部可能会分解成另一种成分（珠光）。因此，液滴运动的精确建模是流体力学中一个非常复杂的问题，在工程中具有许多重要的应用。另一个例子是拥挤区域中个人或汽车的人群运动，通过限制域的部分边界（例如，房间或高速公路）离开；在这些示例中，出口图案很大程度上受到域形状以及出口位置的影响。该项目旨在更好地理解这些问题的特性，并寻求提供一个框架来开发基于计算机的精确过程数值模拟。该项目研究的问题出现在各种物理现象中，包括液体和固体之间的相变、毛细管液滴的运动、拥挤的人群运动和肿瘤生长。特别关注的是从一般初始数据开始的解的渐近行为，无论是在均质化和长期行为的背景下，还是在非线性扩散的“硬压力”极限下。除了偏微分方程中的标准方法（例如积分估计）之外，通常还需要引入几何方法来理解移动界面的逐点行为。该项目的第一部分涉及体积保持几何运动及其长时间行为。挑战在于前沿的合并和分裂可能导致界面的拓扑变化。因此，该区域的大多数结果仅适用于凸面。首席研究员将引入移动平面方法的修改版本来研究更一般的界面类型并研究它们收敛到平衡。第二个子项目涉及准静态近似状态下平坦或倾斜表面上毛细管液滴的演化。虽然在动态毛细管液滴的研究中提出了许多模型，但由于表面粗糙度或液滴体积等参数的行为范围很广，这些模型的分析仍处于早期阶段。这里的目的是解决不同状态下解的适定性和长期行为，对可能的奇点进行分类，并研究参数变化时过程的过渡行为。首席研究员还建议研究集体运动中拥挤区域的出现，例如具有密度约束的人群运动中拥挤区域的演变，或者具有反拥挤压力的癌细胞运动中的肿瘤生长。该计划旨在描述拥挤区域的运动规律，并研究解决方案的稳定性和长期行为。最后，主要研究者建议研究随机介质中的界面均匀化问题，其中不均匀性存在于平流矢量场或潜热中。目标是了解系统中的不均匀性如何与界面的几何形状相互作用，从而影响解决方案的宏观行为。新的调查方法将被引入，使用能源估算和浓度不等式等工具。

A "dynamic free-boundary problem" is the problem of finding the solution of a partial differential equation in a domain whose evolution is a priori unknown. One example is the problem of modeling melting ice, where the interface of ice and water is determined dynamically by the distribution of temperature (i.e., the solution of the heat equation) in the water region. This project addresses some fundamental questions concerning free-boundary problems, such as the existence and long-time behavior of solutions. This is an important topic, since without a proper mathematical theory it is difficult to develop accurate and trustworthy numerical methods for dealing with concrete physical problems. A good example is the problem of liquid drops sliding on a tilted surface. In this case the shape of the drop can change drastically when the velocity is increased, and a singularity (corner) develops at the rear of the drop when the velocity exceeds a certain critical value. At higher speeds the tail of the drop may break into another component (pearling). Accurate modeling of the motion of drops is therefore a highly complex question in fluid mechanics, with many important applications in engineering. Another example is in the crowd motion of individuals or cars in congested areas exiting through part of the boundary of the confining domain (e.g., a room or a highway); in these examples, the exit pattern is heavily influenced by the shape of the domain as well as the location of the exit. The project aims towards a better understanding of the properties of these problems and seeks to provide a framework for developing accurate computer-based numerical simulations of the processes. The problems studied in this project arise in a variety of physical phenomena, including the phase change between liquid and solid, the motion of capillary drops, congested crowd motion, and tumor growth. Particular focus will be on the asymptotic behavior of solutions starting with general initial data, either in the context of homogenization and long-time behavior or in the "stiff-pressure" limit of nonlinear diffusion. In addition to standard methods in partial differential equations, such as integral estimates, it will often be necessary to introduce geometric methods to understand the pointwise behavior of the moving interface. The first part of the project concerns volume-preserving geometric motions and their large-time behavior. The challenge lies in possible topological changes of the interface caused by the merging and splitting of fronts. For this reason most results in this area hold only for convex surfaces. The principal investigator will introduce a modified version of the moving-planes method to investigate a more general class of interfaces and to study their convergence to equilibrium. The second subproject concerns the evolution of capillary drops on a flat or tilted surface in the quasi-static approximation regime. While many models have been proposed in the study of dynamic capillary drops, the analysis of such models is still in its early stages, due the wide range of behavior with respect to parameters such as the roughness of the surface or the volume of the drop. The aim here is to address the well-posedness and long-time behavior of solutions in various regimes, to classify possible singularities, and to investigate the process's transitional behavior upon the change of parameters. The principal investigator also proposes to study the emergence of congested zones in collective motions, for example the evolution of jammed regions in crowd motions with a density constraint, or tumor growth in the motion of cancer cells with anti-crowding pressure. The plan is to characterize the motion law of the congested zones as well as to investigate the stability and long-time behavior of the solutions. Finally, the principal investigator proposes to study interface homogenization problems in random media, where the inhomogeneity is present either in the advection vector field or in the latent heat. The goal is to understand how the inhomogeneities in the system interact with the geometry of the interfaces to affect the macroscopic behavior of the solutions. New approaches will be introduced to the investigation, using tools such as energy estimates and concentration inequalities.