Free boundary problems for capillary surfaces and other nonlinear evolution PDE

毛细管表面和其他非线性演化偏微分方程的自由边界问题

• 批准号: 1201426
• 负责人: Antoine Mellet
• 金额: $ 22.8万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201426&HistoricalAwards=false
• 关键词: Free; boundary; problems; capillary; surfaces

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.

本项目包括三个研究方向。第一部分是研究在模拟液滴在固体载体上的运动时出现的一些偏微分方程（例如，水滴在斜面上滑动）。这部分的研究重点是两个特殊的方程：薄膜方程和准静态近似。这两种模型的主要特征都是存在一条移动的接触线（水滴和固体支撑之间接触区域的边界），其运动是先验未知的。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。主要研究解的存在性问题、解的正则性问题和解的长时间行为问题（特别是解收敛到行波型解）。第二个研究方向涉及某些非局部的三阶抛物方程，特别是在水力裂缝建模中。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）。该项目旨在为这些方程建立一个完整的存在性和正则性理论。尽管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，但由于算子的非局域特性，存在重要的差异。因此，在许多物理上重要的情况下，目前还不知道解的存在。最后一个研究方向是对异常扩散现象的研究。这是由首席研究员发起的一项广泛计划的一部分，该计划旨在研究作为动力学型模型极限而产生的异常扩散机制。他打算推动这个项目研究非谐振子链中的异常热传导。在这样的链中，热量通过振动传递，这种振动可以被模拟为声子气体，其演化可以用玻尔兹曼声子方程来模拟。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。准确地模拟液滴的运动是流体力学中的一个重要问题，在工程上有许多应用。液滴内部流体的运动及其在液滴边缘的行为都涉及到非常复杂的方程，因此人们提出了许多简化模型。这个项目的重点是对其中一些模型进行数学分析，目的是更好地理解它们的基本性质。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的方程。(水力压裂，或“水力压裂”，是通过注入高压流体来扩大岩石裂缝。例如，它参与了页岩气的开采。)该项目解决了有关这些方程的一些基本问题，如解的存在性和规律性。这一点很重要，因为如果没有适当的数学理论，就很难发展出准确可靠的数值方法。因此，该项目将有助于更好地了解这些广泛使用的模型的特性，并为开发精确的基于计算机的数值模拟提供框架。最后，本研究项目包括对学生的培训和指导。事实上，该提案为研究生和本科生提供了许多机会，使他们能够从事具有物理相关应用的无障碍研究项目。