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

9622958马吉达弗拉索夫-泊松方程形成了描述等离子体微观性质的最简单的方程组。因此，这个系统是对等离子体微观性质进行任何预测的起点。弗拉索夫-泊松方程也扮演着另一个非常重要的角色。最近的研究表明，涡量形式的Vlasov-Poisson方程和二维Euler方程之间存在着密切的联系。Vlasov-Poisson方程比Euler方程具有更简单的数学结构，这使得Vlasov-Poisson方程更易于数值分析和求解，并导致许多显式解。因此，Vlasov-Poisson方程形成了一个很好的模型方程系统，可以在考虑任何二维Euler方程的公开问题之前进行研究。作者将考虑Vlasov-Poisson方程的一些基本问题，这些问题与弱解的唯一性和奇性形成有关，也与统计平衡解理论有关。这些问题中有许多(但不是全部)与欧拉方程的未解问题有关。作者还将利用Vlasov-Poisson结果来研究具有涡片初值的欧拉方程的一些基本的公开问题，以激励和指导这项工作。最后，作者将研究关于Vlasov-Poisson方程和Euler方程的数值解的一些有趣的问题。流体力学是研究空气和水等连续介质的科学。科学家通过应用质量、动量和能量守恒等物理学基本定律来解决这一领域的问题。对于大多数问题，这个过程会产生一组太复杂而无法解决的数学方程式。然后，基于物理和/或数学直觉，科学家试图找到原始问题的简化公式，该公式既简单到足以解决，又足够复杂，足以描述原始问题的本质特征。具有不同密度的流体层(如空气和水)或以不同速度运动的流体层之间的混合过程非常复杂，在空气动力学、气象学和海洋学等领域都有广泛的重要应用。例如，飞机上的气流是非常复杂的。为了预测飞机后的尾流或流体中的混合过程，科学家们经常使用一种简化的流体模型，称为涡片。这是对这个问题和许多其他问题的最简单、最现实的描述。不幸的是，尽管涡片模型在流体力学中的许多问题中得到了广泛的应用，但对涡片的许多数学性质仍然知之甚少。这项提议的目的是为许多关于涡片的数学性质的基本问题提供答案。这项拟议工作的成功完成将使人们更好地了解使用涡片模拟复杂流动的优点和局限性。这项工作的起点是描述等离子体的最简单的一组方程，等离子体是一种流体，在其中模拟带电粒子的微观属性。因此，这项研究还应该有助于加深对包含集中电子束的等离子体的性质的理解。

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.