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.

该项目包括三个研究方向。第一个涉及对液滴在固体支撑上运动的建模时出现的一些部分微分方程的研究（例如，水滴向下滑动倾斜的平面）。研究的这一部分侧重于两个特定方程：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（滴和固体支撑之间的接触区域的边界），其运动尚不清楚。因此，这些模型是“自由边界问题”的示例，其数学分析非常具有挑战性。该研究的重点是解决方案的存在问题，其规律性和长期行为（尤其是它们与旅行波型解决方案的融合）。研究的第二方向涉及某些在液压骨折建模时出现的某些非本地，三阶抛物线方程。这些方程让人联想到薄膜方程，但涉及非局部奇异积分算子（例如半拉普拉斯人）。该项目旨在为这些方程开发充分的存在和规律性理论。尽管多年来为薄膜方程式发展的理论的一部分似乎很容易适应该方程，但由于操作员的非局部特征，存在重要的差异。结果，在许多物理上重要的情况下，目前尚不知道解决方案的存在。研究的最后方向涉及对异常扩散现象的研究。这是主要研究者启动的广泛计划的一部分，该计划研究了作为动力学型模型限制的异常扩散方案。他打算推动该程序研究非谐波振荡器链中的异常热传导。在这样的链中，热量是通过可以建模为声子的气体的振动来传输的，该振动是由Boltzmann声子方程建模的。通过研究该方程式的渐近方案，首席研究者试图在热传导中得出非线性异常傅立叶定律。准确地对液滴的运动进行建模是流体力学中的重要问题，具有许多在工程中的应用。物理现象非常复杂（液滴内部的流体运动及其在液滴边缘的行为都涉及非常复杂的方程式），并且已经提出了许多简化的模型。该项目着重于对其中一些模型的数学分析，目的是更好地了解其基本属性。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液压骨折建模时出现的方程式。 （液压压裂或“压裂”，包括通过非常高的压力注入液体来传播岩石裂缝。作为解决方案的存在和规律性。这很重要，因为没有正确的数学理论，很难开发准确且值得信赖的数值方法。因此，该项目将更好地理解这些广泛使用模型的属性，并为开发基于计算机的数值模拟提供了一个框架。最后，该研究计划包括对学生的培训和指导。确实，该建议为研究生和本科生提供了许多机会，可以从事具有与身体相关的应用程序的可访问研究项目。