Nonlinear Partial Differential equations and boundary conditions.

非线性偏微分方程和边界条件。

• 批准号: 1300445
• 负责人: Inwon Kim
• 金额: $ 27.73万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300445&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; equations; boundary

This project includes several aspects of the nonlinear partial differential equations, given in a domain with general geometry. The domain can be given either a priori (fixed boundary problem), or as a part of the problem (free boundary problem). The problems to be discussed arise in a variety of physical phenomena. First we propose to study the averaging behavior for solutions of nonlinear PDEs in a domain with oscillatory boundary data. This type of question arises for instance in the narrow escape problem, which concerns diffusion processes in domains with partly insulated boundaries. We also discuss interface homogenization problems where the inhomogeneity is present, for example, in the perforation structure of the domain (percolation) or in the advection vector field (turbulent flames). Our goal is to understand how the inhomogeneities in the system interact with the geometry of the domain to affect the macroscopic behavior of the solutions. Our second project concerns phase transition phenomena in collective motions, for example the emergence of jammed region in congestion models with a density constraint. We will in particular focus on the characterization of the phase motion law. Lastly we propose to study the evolution of capillary drops on a flat or tilted surface in the quasi-static approximation regime. The aim is to address stability of solutions, classification of possible singularities such as pinning and pinching, and the long-time behavior. For the investigation the tools from probability and PDE theory will be used when they are available.The presence of lower-dimensional structure is ubiquitous in the physical literature, either as a boundary of a domain or a moving interface created by different types of materials. The proposal addresses some fundamental questions concerning these problems, such as existence and long-time behavior of solutions. This is important, since without a proper mathematical theory it is difficult to develop accurate and trustworthy numerical methods. A good example is the problem of sliding drops 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). Accurate modeling of the motion of drops is therefore an important and highly complex question in fluid mechanics, with many 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), where the exit pattern is highly affected by the shape of the domain as well as the location of the exit. The project aim towards a better understanding of the properties of these problems and provide a framework for developing accurate computer-based numerical simulations. Finally, the proposal involves collaboration with students and colleagues at all stages. The plan is to create active environments in applied analysis at the PI's institution by organizing seminars, inviting visitors as well as communicating to experts in related fields, and also by traveling to other institutions, for collaboration and presentations.

这个项目包括几个方面的非线性偏微分方程，在一个域与一般几何。域可以是先验的（固定边界问题），或者作为问题的一部分（自由边界问题）。要讨论的问题出现在各种物理现象中。 首先，我们研究了具有振荡边界数据的区域中非线性偏微分方程解的平均行为。这种类型的问题出现，例如在窄逃逸问题，它涉及的扩散过程中的域与部分绝缘的边界。我们还讨论了界面均匀化问题的不均匀性是存在的，例如，在穿孔结构的域（渗流）或在平流矢量场（湍流火焰）。 我们的目标是了解系统中的不均匀性如何与域的几何形状相互作用，从而影响解的宏观行为。我们的第二个项目关注集体运动中的相变现象，例如在具有密度约束的拥塞模型中拥塞区域的出现。我们将特别集中在相位运动规律的表征。最后，我们建议在准静态近似下研究平坦或倾斜表面上毛细滴的演化。其目的是解决解决方案的稳定性，分类可能的奇点，如钉扎和箍缩，以及长时间的行为。对于调查的工具，从概率和偏微分方程理论将使用时，他们是available.The存在的低维结构是无处不在的物理文献中，无论是作为一个域的边界或移动界面创建的不同类型的材料。该建议解决了一些基本问题，这些问题，如存在性和长期行为的解决方案。这一点很重要，因为没有适当的数学理论，很难开发出准确可靠的数值方法。 一个很好的例子是倾斜表面上的滑滴问题。在这种情况下，当速度增加时，液滴的形状可以急剧变化，并且当速度超过某个临界值时，在液滴的后部形成奇点（拐角）。在较高的速度下，液滴的尾部可能会断裂成另一个组分（珍珠状）。 因此，液滴运动的精确建模是流体力学中一个重要且高度复杂的问题，在工程中有许多应用。另一个例子是拥挤区域中的个人或汽车通过限制域（例如房间或高速公路）的部分边界退出的人群运动，其中退出模式受到域的形状以及出口位置的高度影响。 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一个框架。 最后，该提案涉及与学生和同事在各个阶段的合作。计划通过组织研讨会、邀请来访者以及与相关领域的专家进行交流，并通过前往其他机构进行合作和演示，在PI所在机构的应用分析中创造积极的环境。