Mathematical Sciences: The Incompressible Euler Equations, the Vlasov-Poisson Equations, and Numerical Methods for Their Solutions

数学科学：不可压缩欧拉方程、弗拉索夫-泊松方程及其解的数值方法

• 批准号: 9622958
• 金额: $ 5.7万
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622958&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Incompressible; Euler; Equations

9622958 Majda The Vlasov-Poisson equations form the simplest system of equations which describe the microscopic properties of a plasma. Therefore, this system is the starting point for any prediction about the microscopic properties of a plasma. The Vlasov-Poisson equations also play another very important role. Recent research which involved the proposer demonstrated a close relationship between the Vlasov-Poisson equations and the two dimensional Euler equations in vorticity form. The Vlasov-Poisson equations have a simpler mathematical structure than the Euler equations which makes the Vlasov-Poisson equations easier to analyze and solve numerically, and leads to many explicit solutions. Consequently, the Vlasov-Poisson equations form an excellent system of model equations to study before considering any open problems for the two dimensional Euler equations. The proposer will consider some fundamental problems for the Vlasov-Poisson equations pertaining to the lack of uniqueness and singularity formation for weak solutions and to the theory of statistical equilibrium solutions. Many (but not all) of these problems are related to unsolved problems for the Euler equations. The proposer will also study some fundamental open problems for the Euler equations with vortex sheet initial data, using Vlasov-Poisson results to motivate and guide this work. Finally, the proposer will study some interesting issues regarding the numerical solution of both the Vlasov- Poisson and Euler equations. Fluid dynamics is the study of continuous media like air and water. Scientists formulate problems in this field by applying the fundamental laws of physics such as the conservation of mass, momentum and energy. For most problems this procedure leads to sets of mathematical equations which are too complicated to solve. Based on physical and/or mathematical intuition the scientist then tries to find a simplified formulation of the original problem which is simple enough to solve and sufficiently complicated to describe the essential features of the original problem. The mixing process between layers of fluid which have different densities, like air and water, or layers of fluid moving at different velocities is very complicated and occurs in a wide variety of important applications in fields including aerodynamics, meteorology and oceanography. For example, the flow of air over an airplane is very complicated. In order to predict the trailing wake behind an airplane, or the mixing process in a fluid, scientists often use a simplified model of a fluid called a vortex sheet. This is the simplest realistic description of this and many other problems. Unfortunately, despite the wide-spread use of vortex sheet models in many problems in fluid dynamics, many mathematical properties of vortex sheets remain poorly understood. The goal of this proposal is to provide answers to many of the fundamental questions about the mathematical properties of vortex sheets. Successful completion of the proposed work will lead to an increased understanding of the strengths and limitations when using vortex sheets to model complicated flows. The starting point for this work is the simplest set of equations which describe a plasma, a fluid in which the microscopic properties of charged particles are modeled. Therefore, this study should also lead to an increased understanding of the properties of a plasma containing a concentrated beam of electrons.

小行星9622958 Vlasov-Poisson方程是最简单的方程组， 描述了等离子体的微观特性。 因此，该系统是 任何关于一种物质的微观性质的预测的出发点 等离子体 Vlasov-Poisson方程也是另一个非常重要的 作用最近的一项研究表明， Vlasov-Poisson方程与二维 涡量形式的欧拉方程。 Vlasov-Poisson方程有一个 比欧拉方程更简单的数学结构， Vlasov-Poisson方程更容易分析和数值求解，并导致 许多明确的解决方案。 因此，弗拉索夫-泊松方程 形成一个优秀的模型方程系统，在考虑之前进行研究 二维欧拉方程的任何开放问题。 要保人 将考虑Vlasov-Poisson方程的一些基本问题 关于缺乏唯一性和奇点形成的弱 解和统计平衡解的理论。 许多（但 并非所有的这些问题都与欧拉方程的未解问题有关 方程 提案人还将研究一些基本的开放问题 对于具有涡面初始数据的Euler方程，使用Vlasov-Poisson 以激励和指导这项工作。 最后，提案人将研究 关于Vlasov-Poisson和Euler方程数值解的一些有趣的问题。 流体动力学是研究连续介质如空气和水的学科。 科学家们通过应用基本定律， 物理学的基本原理，如质量守恒，动量守恒和能量守恒。 为 大多数问题，这一过程导致了一组数学方程， 太过复杂而无法解决 基于物理和/或数学 直觉，科学家然后试图找到一个简化的公式， 原始问题，它足够简单，可以解决， 复杂的描述原始问题的本质特征。 的 具有不同密度的流体层之间的混合过程，如 空气和水，或以不同速度移动的流体层， 复杂，并出现在各种重要的应用领域， 包括空气动力学、气象学和海洋学。 比如说 飞机上方的空气流动非常复杂。 为了预测 科学家们认为， 通常使用一种叫做涡面的流体简化模型。 这是 最简单的现实描述这个问题和许多其他问题。 不幸的是，尽管涡面模型在许多领域得到了广泛的应用， 流体动力学中的问题，涡面的许多数学性质 仍然知之甚少。 本提案的目的是提供答案 许多关于数学性质的基本问题， 涡面 成功完成拟议的工作将导致 增加对使用Vortex时的优势和局限性的了解 用于模拟复杂的流动。 这项工作的出发点是 最简单的一组方程，描述了等离子体，一种流体， 带电粒子的微观特性被建模。 因此本 研究还应导致增加了解的性质， 含有集中的电子束的等离子体。