Interfaces, Degenerate Partial Differential Equations, and Convexity

接口、简并偏微分方程和凸性

• 批准号: 2348846
• 负责人: Benjamin Weinkove
• 金额: $ 14.56万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348846&HistoricalAwards=false
• 关键词: Interfaces; Degenerate; Partial; Differential; Equations

Partial differential equations (PDE) are essential mathematical objects for modeling physical processes. This project aims to understand the properties of some fundamental PDE models for the diffusion of gas, the shape of liquid droplets, and electric transmission in composite materials. Many such processes exhibit an interface, where the equation becomes degenerate or singular. In the case of a model of gas diffusion through a porous medium, the interface is the set which separates the region where there is gas from the region where there is no gas. The boundary of a liquid droplet is another example of an interface. The project will investigate qualitative and quantitative properties of these interfaces, including convexity and smoothness. For a composite material, consisting of two materials with different conductivity properties, the interface is where these materials meet. The Principal Investigator (PI) will study the behavior of the electric field when one of the materials has a very thin part. This project has possible implications for material failure, an important question in Engineering. Students and postdoctoral scholars will be trained on the techniques and theory of these PDE models.This project centers on four topics, connected by the themes of interfaces, degeneracies, and convexity/concavity. The porous medium equation is a nonlinear degenerate parabolic equation used to model the diffusion of gas. The PI will investigate questions of concavity and convexity of solutions and finding global optimal regularity estimates. Secondly, the PI will study linear PDE whose coefficients are discontinuous along two almost touching interfaces, a model for transmission problems and composite materials. In this setting, the PI will investigate new approaches to obtaining optimal gradient estimates in the thin region between the interfaces. A third project is to study linear equations which are parabolic on the interior on a fixed domain but are degenerate at the boundary. These equations arise as linearizations of the porous medium equation and the Gauss curvature flow. The PI will investigate optimal conditions for existence and uniqueness of smooth solutions. Finally, questions of concavity of solutions to the elliptic torsion problem with Dirichlet boundary conditions and the dynamic version of this equation, known as the quasi-static droplet model, will be studied. In addition, the PI will carry out summer research projects with undergraduates, exploring explicit solutions to these equations.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.

部分微分方程（PDE）是用于建模物理过程的必不可少的数学对象。该项目旨在了解某些基本PDE模型的特性，以扩散气体，液滴的形状以及复合材料中的电动传输。许多这样的过程都表现出一个界面，其中方程变成归纳或单数。在通过多孔介质扩散的气体扩散模型的情况下，界面是将气体与没有气体区域的区域分开的集合。液滴的边界是界面的另一个例子。该项目将研究这些接口的定性和定量特性，包括凸度和光滑度。对于由具有不同电导率特性的两种材料组成的复合材料，界面是这些材料相遇的地方。当其中一种材料的部分很薄时，主要研究者（PI）将研究电场的行为。该项目可能对材料故障产生影响，这是工程学中的重要问题。学生和博士后学者将接受这些PDE模型的技术和理论的培训。该项目中心以四个主题为中心，该主题由接口，变性和凸度/凹陷的主题相连。多孔培养基方程是用于模拟气体扩散的非线性退化抛物线方程。 PI将研究解决方案的凹度和凸状态的问题，并找到全球最佳的规律性估计。其次，PI将研究线性PDE，其系数沿两个几乎接触的接口是不连续的，这是传输问题和复合材料的模型。在这种情况下，PI将研究在接口之间薄区域中获得最佳梯度估计值的新方法。第三个项目是研究固定结构域上内部抛物线的线性方程，但在边界处是堕落的。这些方程是作为多孔培养基方程和高斯曲率流的线性化产生的。 PI将研究平滑溶液的存在和独特性的最佳条件。最后，将研究对椭圆扭转问题的解决方案的问题，并研究该方程式的动态版本（称为准静态液滴模型）。此外，PI将在本科生中进行夏季研究项目，探讨这些方程式的明确解决方案。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来支持的。