Topics in Applied Partial Differential Equations

应用偏微分方程主题

• 批准号: 1104415
• 负责人: Alexander Kiselev
• 金额: $ 33万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104415&HistoricalAwards=false
• 关键词: Topics; Applied; Partial; Differential; Equations

The project covers several topics. The first is the development of new methods for analysis of active scalar equations. These are nonlinear and nonlocal partial differential equations that, in particular, include classical two-dimensional Euler equation describing ideal fluid flow, and surface quasi-geostrophic equation arising in atmospheric science. Active scalars have been used to model a wide range of phenomena in nature, including formation of fronts in atmosphere, diffusion in porous medium and evolution of vortex sheets. Research will build on recent work of the principal investigator (PI) by finding more general maximum principles (bounds controlling solutions). A novel aspect of this research is that these bounds are nonlocal, which may be just the right fit for nonlocal equations like active scalars. New techniques are expected to provide progress on key questions in mathematical fluid mechanics involving structure of solutions to active scalar equations, their regularity and possible singularity formation. The second direction focuses on enhancement of diffusion by fluid flow. Numerous processes in nature and engineering, starting from nuclear burning in stars to combustion in engines to reactions in living organisms depend on this phenomenon. The goal is, building on the earlier research of the PI, to improve understanding of flows that are most efficient in speeding up diffusion and mixing. The problem is at the interface of partial differential equations, dynamical systems and Fourier analysis. The methods to be developed here will be relevant in a more general context. They apply in problems that involve convergence to equilibrium in systems that contain both dissipative and fast unitary parts of dynamics. The third direction, biomixing, is motivated by a problem of coral broadcast spawning that has been communicated to the PI by an oceanographer. It involves studying improvement of the efficiency of biological reactions by chemotaxis. A model is proposed that adds chemotaxis term to the previously studied models of the process. The effect chemotaxis has on fertilization rate is likely to be crucial for health of many biosystems. The goal of this direction of the project will be to better understand coral spawning process and quantify an important role chemotaxis plays in achieving the reproduction success.The project focuses on several problems in fluid mechanics. In one direction, novel techniques are developed that will provide insight into behavior of solutions to equations modeling diverse phenomena from temperature evolution in the atmosphere to traffic flow dynamics. In other direction, the problem of mixing in fluid flow will be studied, and classes of flows that are most efficient mixers will be identified. The question of efficient mixing is of interest in many industries, including food processing and chemical engineering. Another direction of the project research improves reproduction models for a class of marine animals including corals. Coral atolls are important ecosystems that are under stress worldwide due to climate change and pollution. The models that will be developed in the project are important for better understanding of coral life cycle, and will be of interest in oceanography and ecology. The project has a significant and broad training component, and will involve a postdoc, graduate and undergraduate students working on problems related to the project research.

该项目涵盖几个主题。第一个是发展新的方法来分析有源标量方程。这些是非线性和非局部偏微分方程，特别是，包括经典的二维欧拉方程描述理想的流体流动，并在大气科学中出现的地面准地转方程。主动标量已被广泛用于模拟自然界中的许多现象，包括大气锋面的形成、多孔介质中的扩散以及涡面的演化。研究将建立在主要研究者（PI）的最新工作基础上，找到更一般的最大值原理（边界控制解决方案）。这项研究的一个新的方面是，这些边界是非局部的，这可能是正确的适合非局部方程，如积极标量。新技术预计将在数学流体力学的关键问题上取得进展，这些问题涉及活动标量方程解的结构、其规律性和可能的奇异性形成。第二个方向侧重于通过流体流动增强扩散。自然界和工程中的许多过程，从恒星的核燃烧到发动机的燃烧，再到生物体的反应，都依赖于这种现象。我们的目标是，在PI早期研究的基础上，提高对加速扩散和混合最有效的流动的理解。这个问题是在偏微分方程，动力系统和傅立叶分析的接口。这里要发展的方法将在更一般的情况下是相关的。它们适用于包含耗散和快速单一动力学部分的系统中涉及收敛到平衡的问题。第三个方向，biomixing，是由珊瑚广播产卵的问题，已传达给PI的海洋学家的动机。它涉及研究通过趋化性提高生物反应的效率。提出了一个模型，增加了趋化性长期到以前研究的模型的过程。趋化性对受精率的影响可能对许多生物系统的健康至关重要。该项目的目标是更好地了解珊瑚产卵过程，并量化趋化性在实现繁殖成功中的重要作用。该项目侧重于流体力学中的几个问题。在一个方向上，开发新的技术，将提供洞察的行为的解决方案的方程建模不同的现象，从大气中的温度演变的交通流动力学。在另一个方向上，将研究流体流动中的混合问题，并确定最有效的混合器的流动类别。有效混合的问题在许多行业都很重要，包括食品加工和化学工程。该项目研究的另一个方向是改进包括珊瑚在内的一类海洋动物的繁殖模型。珊瑚环礁是重要的生态系统，由于气候变化和污染，全世界都受到压力。该项目将开发的模型对于更好地了解珊瑚的生命周期非常重要，并将在海洋学和生态学中引起兴趣。该项目有一个重要的和广泛的培训组成部分，并将涉及博士后，研究生和本科生的工作与项目研究的问题。