Topics in Applied Partial Differential Equations

该项目涵盖了几个主题。首先是发展了分析有源标量方程的新方法。这些是非线性和非局部偏微分方程，特别是包括描述理想流体流动的经典二维欧拉方程和大气科学中出现的地表准地转方程。主动标量已被用于模拟广泛的自然现象，包括大气锋面的形成、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）最近工作的基础上，通过寻找更一般的最大原理（边界控制解）。本研究的一个新颖之处在于这些边界是非局部的，这可能正好适合于像活动标量这样的非局部方程。新技术有望在数学流体力学的关键问题上取得进展，包括主动标量方程解的结构、它们的规律性和可能的奇点形成。第二个方向侧重于流体流动对扩散的增强。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，在早期PI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。问题是在偏微分方程，动力系统和傅立叶分析的界面。在此拟定的方法将适用于更一般的情况。它们适用于包含耗散和快速统一动力部分的系统中涉及收敛到平衡的问题。第三个方向，生物混合，是由一个海洋学家向PI传达的珊瑚广播产卵问题引起的。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，将趋化性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方面所起的重要作用。这个项目集中研究流体力学中的几个问题。在一个方向上，新技术的发展将提供对从大气温度演变到交通流动力学的各种现象建模的方程解的行为的见解。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流的类别。有效混合的问题是许多行业感兴趣的，包括食品加工和化学工程。该项目的另一个研究方向是改善包括珊瑚在内的一类海洋动物的繁殖模型。由于气候变化和污染，珊瑚环礁是全球范围内面临压力的重要生态系统。该项目将开发的模型对更好地了解珊瑚的生命周期很重要，对海洋学和生态学也很有意义。该项目有一个重要而广泛的培训组成部分，将涉及博士后、研究生和本科生，他们将从事与项目研究相关的问题。

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.