Topics in Applied Partial Differential Equations

该项目涵盖了几个主题。一是发展分析有源标量方程的新方法。这些方程是非线性的非局部偏微分方程组，特别是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地表准地转方程。活动标量已被用来模拟自然界中的一系列现象，包括大气中锋面的形成、多孔介质中的扩散和涡旋片的演化。研究将建立在首席研究员(PI)最近的工作基础上，通过寻找更一般的最大值原理(界控制解)。这项研究的一个新的方面是，这些界限是非局部的，这可能正好适合于像活动标量这样的非局部方程。新的技术有望在数学流体力学的关键问题上取得进展，这些问题涉及活动标量方程的解的结构、它们的正则性和可能的奇性形成。第二个方向侧重于流体流动对扩散的强化作用。自然界和工程学中的许多过程，从恒星中的核燃烧到发动机中的燃烧，再到生物体中的反应，都依赖于这一现象。其目标是在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交界处。这里要开发的方法将在更广泛的背景下相关。它们适用于涉及收敛到平衡的问题，这些系统既包含耗散的动力学部分，也包含快速的么正部分动力学。第三个方向，生物融合，是由珊瑚传播产卵的问题推动的，这个问题已经由一名海洋学家传达给了PI。它包括研究通过趋化作用提高生物反应的效率。在已有的过程模型的基础上增加了趋化性项，提出了一个模型。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。该项目这一方向的目标是更好地了解珊瑚产卵过程，并量化趋化性在成功繁殖中所起的重要作用。该项目重点研究流体力学中的几个问题。在一个方向上，开发了新的技术，这些技术将提供对建模从大气温度演变到交通流动力学的各种现象的方程的解的行为的洞察。在另一个方向，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。高效混合的问题在许多行业都很有意义，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。珊瑚礁是世界范围内因气候变化和污染而面临压力的重要生态系统。将在该项目中开发的模型对于更好地了解珊瑚的生命周期非常重要，并将对海洋学和生态学产生兴趣。该项目有一个重要而广泛的培训部分，将有博士后、研究生和本科生参与研究与项目研究有关的问题。

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.