Free boundary problems for capillary surfaces and other nonlinear evolution PDE

该项目包括三个研究方向。第一个涉及在固体支持物上的液滴运动的建模中出现的一些偏微分方程的研究（例如，沿着斜面向下滑动的水滴）。这部分的研究集中在两个特殊的方程：薄膜方程和准静态近似。这两种模型的主要特征是存在移动接触线（液滴和固体支撑物之间的接触区域的边界），其运动先验未知。因此，这些模型是“自由边界问题”的例子，其数学分析非常具有挑战性。研究的重点是解的存在性，它们的规律性，以及它们的长时间行为（特别是它们收敛到行波型解）。研究的第二个方向涉及某些非局部的，三阶抛物方程，特别是在水力压裂的建模。这些方程让人想起薄膜方程，但涉及非局部奇异积分算子（如半拉普拉斯算子）。该项目旨在为此类方程开发一个完整的存在性和正则性理论。虽然多年来为薄膜方程发展的部分理论似乎很容易适应这个方程，但由于算子的非局部性质，存在重要的差异。因此，在许多物理上重要的情况下，解的存在性目前还不清楚。最后一个研究方向是反常扩散现象的研究。这是首席研究员发起的一个广泛计划的一部分，该计划旨在研究由于动力学类型模型的限制而产生的异常扩散机制。他打算推动这一计划，以研究非谐振子链中的反常热传导。在这样的链中，热通过振动传输，振动可以被建模为声子气体，其演化由玻尔兹曼声子方程建模。通过研究该方程的渐近状态，主要研究者试图推导出热传导的非线性反常傅立叶定律。精确模拟液滴的运动是流体力学中的一个重要问题，在工程中有许多应用。物理现象极其复杂（液滴内部流体的运动及其在液滴边缘的行为都涉及非常复杂的方程），已经提出了许多简化模型。该项目侧重于对其中一些模型进行数学分析，目的是更好地了解其基本特性。最终，目标是将这些属性与实验进行比较，以验证（或无效）各种模型。该项目的另一个方面涉及水力压裂建模中出现的方程。（水力压裂，或“压裂”，包括通过注入具有非常高压力的流体来扩展岩石裂缝。例如，它参与了页岩气的开采。）该项目解决了关于这些方程的一些基本问题，如解的存在性和正则性。这一点很重要，因为没有适当的数学理论，很难发展出精确可靠的数值方法。因此，该项目将导致更好地了解这些广泛使用的模型的属性，并提供一个框架，以开发精确的计算机为基础的数值模拟。最后，这项研究计划包括对学生的培训和指导。事实上，这一提议为研究生和本科生提供了许多机会，使他们能够从事与物理相关的研究项目。

This project includes three directions of research. The first concerns the study of some partial differential equations that arise in the modeling of the motion of liquid droplets on a solid support (e.g., a water drop sliding down an inclined plane). This part of the research focuses on two particular equations: the thin film equation and the quasi-static approximation. The main feature of both of these models is the presence of a moving contact line (the boundary of the contact region between the drop and the solid support) whose motion is not known a priori. These models are thus examples of "free boundary problems," whose mathematical analysis is very challenging. The research focuses on the questions of existence of solutions, their regularity, and their long-time behavior (especially their convergence to traveling-wave-type solutions). The second direction of research concerns certain nonlocal, third-order parabolic equations that arise, in particular, in the modeling of hydraulic fractures. These equations are reminiscent of the thin film equation, but involve nonlocal singular integral operators (such as the half-Laplacian). The project aims at developing a full existence and regularity theory for such equations. Though parts of the theory developed over the years for the thin film equation seem to adapt readily to this equation, there are important differences due to the nonlocal character of the operator. As a consequence, the existence of solutions is not presently known in many physically important cases. The last direction of research concerns the study of anomalous diffusion phenomena. This is part of a broad program initiated by the principal investigator to study anomalous diffusion regimes arising as limits of kinetic-type models. He intends to push this program to study anomalous heat conduction in chains of anharmonic oscillators. In such chains, heat is transported by vibrations that can be modeled as a gas of phonons, whose evolution is modeled by the Boltzmann phonon equation. By studying asymptotic regimes for this equation, the principal investigator seeks to derive a nonlinear anomalous Fourier law for heat conduction.Accurately modeling the motion of liquid droplets is an important problem in fluid mechanics with many applications in engineering. The physical phenomena are extremely complex (the motion of the fluid inside the droplet and its behavior at the edge of the droplet both involve very complicated equations), and many simplified models have been proposed. This project focuses on the mathematical analysis of some of those models with the aim of better understanding their fundamental properties. Ultimately, the goal is to compare these properties with experiments to validate (or invalidate) the various models. Another aspect of the project involves equations that arise in the modeling of hydraulic fracture. (Hydraulic fracturing, or "fracking," consists in propagating rock fractures by the injection of fluids with very high pressure. It is involved, for instance, in the extraction of shale gas.) The project addresses some fundamental questions concerning these equations, such as the existence and regularity of solutions. This is important, since without a proper mathematical theory it is very difficult to develop accurate and trustworthy numerical methods. The project will thus lead to a better understanding of the properties of these widely used models and provide a framework for developing accurate computer-based numerical simulations. Finally, this research program includes the training and mentoring of students. Indeed, this proposal offers many opportunities for both graduate and undergraduate students to work on accessible research projects with physically relevant applications.