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.

该项目涵盖多个主题。首先是开发主动标量方程分析的新方法。这些是非线性和非局部偏微分方程，特别包括描述理想流体流动的经典二维欧拉方程，以及大气科学中出现的表面准地转方程。主动标量已被用来模拟自然界中的各种现象，包括大气中锋面的形成、多孔介质中的扩散以及涡旋片的演化。研究将建立在首席研究员 (PI) 最近的工作基础上，寻找更通用的最大原理（边界控制解决方案）。这项研究的一个新颖之处是这些界限是非局部的，这可能正好适合像主动标量这样的非局部方程。新技术有望在数学流体力学的关键问题上取得进展，这些问题涉及主动标量方程的解的结构、它们的规律性和可能的​​奇点形成。第二个方向侧重于通过流体流动增强扩散。自然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。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。该问题存在于偏微分方程、动力系统和傅立叶分析的交叉点上。这里要开发的方法将在更一般的背景下相关。它们适用于涉及包含动力学耗散部分和快速酉部分的系统中收敛到平衡的问题。第三个方向是生物混合，其动机是由海洋学家向 PI 传达的珊瑚广播产卵问题。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，将趋化项添加到先前研究的过程模型中。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。该项目这个方向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所起的重要作用。该项目重点关注流体力学中的几个问题。在一个方向上，新技术的开发将提供对模拟从大气温度演变到交通流动力学等各种现象的方程解的行为的洞察。在其他方向上，将研究流体流动中的混合问题，并且将识别最有效混合器的流动类别。有效混合的问题引起了许多行业的关注，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，全球范围内都面临着压力。该项目将开发的模型对于更好地了解珊瑚生命周期非常重要，并且将引起海洋学和生态学的兴趣。该项目具有重要而广泛的培训内容，将涉及博士后、研究生和本科生，致力于解决与项目研究相关的问题。