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

该项目旨在增进流体数学知识，流体数学是一门与工程、物理、化学和许多其他科学相关的学科。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。该项目研究将解决流体动力学经典方程的基本性质、解的定性性质以及建模应用。该项目重点关注三个主要方向。第一组问题涉及全局规律性与有限时间爆炸，将针对一系列流体力学基本方程进行研究。将考虑 3D 欧拉和纳维-斯托克斯的轴对称解，研究新的潜在奇异场景并寻求新的正则性标准。主动标量，例如来自大气科学的表面准地转方程，也将被分析——这里的工作将集中在研究解的性质、寻找新的李亚普诺夫泛函和新颖的规律性估计。在第二个方向上，我们将寻求有关长期动态的更详细信息。这将包括对二维欧拉方程的一些长期开放猜想的研究，包括逆能量级联、混合和小尺度形成的可能机制。我们还将研究这种情况下的被动标量模型和流的混合属性。还将研究趋化性生物混合，着眼于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 - 流体中颗粒的微观结构变得很重要。可以通过向流体动力学系统添加动力学方程并引入物理自然耦合来考虑颗粒之间的形状和相互作用。对此类系统的解决方案、其规律性和定性特性的分析将成为项目工作的一部分。流体在自然、科学和工程中无处不在。涉及流体的各种现象出现在大气和海洋科学、天体物理学、化学和生物学中，并通过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。它们描述了广泛的复杂现象，是非线性的，并且通常是非局部的。由于其复杂性，即使是 3D 欧拉和纳维-斯托克斯等经典方程也远未得到很好的理解。所提出的研究涉及数学的几个核心领域——偏微分方程、动力系统、泛函分析和傅立叶分析。这个 FRG 项目汇集了几位处于数学流体力学最新发展前沿的研究人员。不同的参与者给项目带来不同的优势。预计 FRG 框架内的密切合作将带来新思想和新方法的发展，并导致流体数学领域的活动爆发。开发的新技术和工具可能会对数学、生物学和大气科学等邻近领域产生影响。 FRG 活动的一个重要部分是对初级研究人员的培训。 流体力学数学涵盖了广泛的有效技术，这些技术适用于流体之外的领域。培训活动将包括一个暑期学校、两个研讨会、小组会议、课程开发、研究研讨会和针对高级本科生的研究项目。主要研究者将对各项培训活动进行广泛宣传，努力招募有才华、积极进取、多元化的学员。将特别关注数学领域代表性不足的群体的招募。

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.