Dynamic Free Boundary Problems

动态自由边界问题是指在演化先验未知的域中求解偏微分方程。一个例子是冰融化问题，其中冰和水的界面是由水区域的温度分布或热方程的解动态确定的。特别关注那些没有规律性给出解决方案的问题。 一个很好的例子是水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）。另一个例子是，当最初将水（较重的流体）倒在油的顶部时，油滴向上移向容器中的水。该项目旨在更好地理解这些问题的性质，并为开发精确的基于计算机的数值模拟提供框架。研究生将通过参与研究来接受培训。要讨论的问题出现在各种物理现象中，包括毛细管液滴的运动、人群运动、肿瘤生长以及多孔介质中的流体过滤。低维结构的存在在物理文献中无处不在，无论是作为域的边界还是进化的单一部分。许多原本很容易理解的非线性偏微分方程问题在与移动界面耦合时面临着重大挑战，即使是在看似简单的设置中。除了问题的非线性之外，困难还在于问题的非局部性，即解的行为取决于自由边界的全局几何形状。由于缺乏自由边界的先验规律性，动态问题面临着额外的困难。通过研究提出的问题，我们的目标是开发解决这些困难的通用方法。除了能量和积分估计之外，通常还需要引入几何方法来理解移动界面的逐点行为。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamic free boundary problem refers to solving partial differential equations in a domain whose evolution is a priori unknown. One example is the problem of melting ice, where the interface of ice and water is determined dynamically by the distribution of temperature, or solution of the heat equation, in the water region. A particular focus is given on problems where solutions are given with very little regularity. A good example is the problem of drops sliding on 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 exceed a certain critical value. At higher speeds the tail of the drop may break into another component (pearling). Another example is oil drops moving up towards water in a container when initially water, the heavier fluid, is poured on top of oil. The project aims towards a better understanding of the properties of these problems and provide a framework for developing accurate computer-based numerical simulations. Graduate students will be trained by participating in the research.The problems to be discussed arise in a variety of physical phenomena, including the motion of capillary drops, crowd motion, tumor growth, and filtration of fluids in porous media. The presence of lower-dimensional structure is ubiquitous in the physical literature, either as a boundary of a domain or a singular part of an evolution. Many nonlinear PDE problems which are otherwise well-understood face significant challenges when they are coupled with a moving interface, even in seemingly simple settings. Besides the nonlinearity of the problem, the difficulty lies in the non-locality of the problem, in the sense that the behavior of solutions depends on the global geometry of the free boundary. Dynamic problems face an additional difficulty coming from the lack of a priori regularity of the free boundaries. By studying proposed problems we aim to develop general methods to deal with such difficulties. In addition to energy and integral estimates, it will be often necessary to introduce geometric methods to understand pointwise behavior of the moving interface.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.