Dynamic Free-Boundary Problems

“动态自由边界问题”是在一个演化过程先验未知的区域内求解偏微分方程的问题。一个例子是模拟融化冰的问题，其中冰和水的界面是由水区域的温度分布（即热方程的解）动态决定的。本课题解决了自由边界问题的一些基本问题，如解的存在性和长期行为。这是一个重要的课题，因为没有适当的数学理论，很难发展出准确可靠的数值方法来处理具体的物理问题。一个很好的例子就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，液滴的形状会发生剧烈变化，当速度超过某个临界值时，液滴后部会出现一个奇点（角落）。在较高的速度下，水滴的尾部可能会破裂成另一种成分（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，在工程上有许多重要的应用。另一个例子是在拥挤地区的个人或汽车的人群运动中，通过部分限制域的边界（例如，房间或高速公路）；在这些示例中，出口模式受到域的形状以及出口位置的严重影响。该项目旨在更好地了解这些问题的性质，并寻求为开发精确的基于计算机的过程数值模拟提供框架。本课题研究的问题产生于各种物理现象，包括液体与固体的相变、毛细液滴的运动、拥挤人群的运动、肿瘤的生长。特别关注从一般初始数据开始的解的渐近行为，无论是在均匀化和长时间行为的背景下，还是在非线性扩散的“刚性压力”极限下。除了偏微分方程中的标准方法，如积分估计，通常有必要引入几何方法来理解移动界面的逐点行为。项目的第一部分涉及保持体积的几何运动及其大时间行为。挑战在于锋面的合并和分裂可能导致界面的拓扑变化。由于这个原因，这个领域的大多数结果只适用于凸面。首席研究员将引入移动平面方法的改进版本，以研究更一般的界面类别并研究其收敛到平衡状态。第二个子项目涉及在准静态近似制度下在平坦或倾斜表面上毛细液滴的演变。虽然在动态毛细液滴的研究中已经提出了许多模型，但由于表面粗糙度或液滴体积等参数的行为范围很广，对这些模型的分析仍处于早期阶段。这里的目的是解决在各种情况下的解的适定性和长时间行为，分类可能的奇点，并研究过程的过渡行为在参数的变化。首席研究员还建议研究集体运动中拥挤区域的出现，例如在密度限制下人群运动中拥挤区域的演变，或者在抗拥挤压力下癌细胞运动中的肿瘤生长。该计划旨在描述拥堵区域的运动规律，并研究解决方案的稳定性和长期行为。最后，作者建议研究随机介质中的界面均匀化问题，其中平流矢量场或潜热中存在不均匀性。目标是了解系统中的非均质性如何与界面的几何形状相互作用，从而影响解的宏观行为。新的方法将被引入到调查中，使用诸如能量估计和集中不平等等工具。

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.