FRG: Collaborative Research: Singularities, mixing and long time behavior in nonlinear evolution

FRG：协作研究：非线性演化中的奇异性、混合和长期行为

• 批准号: 1159138
• 负责人: Thomas Hou
• 金额: $ 25万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1159138&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Singularities; mixing

The project seeks to advance knowledge in mathematics of fluids, a subjectwith links to engineering, physics, chemistry and many other sciences. The major goal of the project is the development of new techniques to achieve breakthroughs in our understanding of fluid dynamics phenomena. The project research will address fundamental properties of the classical equations of fluid dynamics, qualitative properties of solutions, and modeling applications. The project focuses on three main directions. The first set of problems concerns global regularity vs finite time blow up that will be investigated for a range of fundamental equations of fluid mechanics. Axi-symmetric solutions for 3D Euler and Navier-Stokes will be considered, new potentially singular scenarios will be studied and new regularity criteria will be sought. Active scalars, such as surface quasi-geostrophic equation coming from atmospheric science, will also be analyzed -- here the effort will concentrate on studying properties of solutions, search for new Lyapunov functionals and novel regularity estimates. In the second direction, we will seek more detailed information on long time dynamics. This will include research on some long-open conjectures for 2D Euler equation, including possible mechanisms of inverse energy cascade, mixing and small scales formation. We will also work on passive scalar models and mixing properties of flows in this context. Biomixing by chemotaxis will be investigated as well, with an eye towards applications in ecology and marine biology. The third direction focuses on complex fluid models. In many applications - for instance, in studies of particle suspensions or solutions - the microscopic structure of particles in the fluid becomes important. The shape and interactions between the particles can be taken into account by adding kinetic equations to the fluid dynamics systems and introducing physically natural couplings. Analysis of solutions to such systems, their regularity and qualitative properties will be a part of the project work.Fluids are ubiquitous in nature, science and engineering. Diverse phenomena involving fluids appear in atmospheric and ocean science, astrophysics, chemistry and biology, and are described by partial differential equations of fluid mechanics. These equations are some of the most difficult partial differential equations to analyze. They describe a wide range of complex phenomena, are nonlinear, and usually nonlocal. Due to their complexity, even the classical equations such as 3D Euler and Navier-Stokes are far from well understood. The proposed research lies at the interface of several central areas of mathematics - partial differential equations, dynamical systems, functional analysis and Fourier analysis. This FRG project brings together several researchers that have been at the forefront of recent developments in mathematical fluid mechanics. Different participants bring different strengths to the project. It is expected that intensive collaboration within FRG framework will lead to development of new ideas and approaches and result in a burst of activity in mathematics of fluids. New techniques and tools developed are likely to have an impact in neighboring areas of mathematics, biology, and atmospheric science. An important part of the FRG activity will be training of junior researchers. Mathematics of fluid mechanics covers a broad range of effective techniques, which are applicable beyond fluids. The training activities will include a summer school, two workshops, group meetings, course development, research seminars and research projects for advanced undergraduate students. The principal investigators will advertise all training activities broadly, and strive to recruit talented, motivated, and diverse trainees. Special attention will be paid to recruitment of groups under represented in mathematics.

该项目旨在推进流体数学的知识，这是一个与工程，物理，化学和许多其他科学相关的主题。该项目的主要目标是开发新技术，以实现我们对流体动力学现象的理解的突破。该项目的研究将解决经典流体动力学方程的基本性质，解决方案的定性性质，以及建模应用。该项目侧重于三个主要方向。第一组问题涉及全局正则性与有限时间爆破，将研究流体力学的一系列基本方程。将考虑三维欧拉和纳维尔-斯托克斯方程的轴对称解，将研究新的潜在奇异情况，并寻求新的正则性准则。活动标量，如来自大气科学的表面准地转方程，也将进行分析-在这里的努力将集中在研究解决方案的性质，寻找新的李雅普诺夫泛函和新的规律性估计。在第二个方向，我们将寻求更详细的长期动态信息。这将包括一些长期开放的二维欧拉方程的解的研究，包括逆能量级联，混合和小尺度形成的可能机制。我们还将在这方面的被动标量模型和混合特性的流动。通过趋化性的生物混合也将被研究，着眼于在生态学和海洋生物学中的应用。第三个方向侧重于复杂的流体模型。在许多应用中，例如，在颗粒悬浮液或溶液的研究中， 流体中的颗粒变得很重要。颗粒之间的形状和相互作用可以通过将动力学方程添加到流体动力学系统并引入物理自然耦合来考虑。分析这些系统的解，它们的规律性和定性性质将是项目工作的一部分。流体在自然界、科学和工程中无处不在。在大气和海洋科学、天体物理学、化学和生物学中，涉及流体的各种现象都可以用流体力学的偏微分方程来描述。这些方程是一些最难分析的偏微分方程。他们 描述了广泛的复杂现象，是非线性的，通常 非本地的由于其复杂性，即使是经典方程，例如 因为三维欧拉和纳维尔-斯托克斯方程还远未被很好地理解。的 拟议的研究在于数学的几个中心领域的接口-偏微分方程，动力系统，泛函分析和傅立叶分析。这个联邦德国项目 汇集了几个研究人员一直在最前沿的数学流体力学的发展。不同 参与者为项目带来不同的优势。预计联邦德国框架内的密切合作将导致 新的思想和方法的发展，并导致在流体数学的活动爆发。开发的新技术和工具 可能会对数学、生物学和大气科学的邻近领域产生影响。联邦德国的重要组成部分 活动将是对初级研究人员的培训。 流体力学的数学涵盖了广泛的有效技术， 适用于流体之外。培训活动将包括 暑期学校，两个讲习班，小组会议，课程开发， 为高年级本科生举办的研究研讨会和研究项目 学生主要研究人员将宣传所有培训 活动广泛，并努力招募有才华，积极主动， 不同的学员。将特别注意征聘 在数学中表现不足的群体。