Mathematical Sciences: A Study of Weak Solutions for the Euler Equations and Related Equations from Plasma Physics

小行星9307726 关于不可压Euler方程的弱解，还有许多基本问题没有解决，如涡面问题，以及通过平滑初值或增加粘性对这些弱解进行正则化的问题。 此外，由于这些问题是非常具有挑战性的，试图用数值方法来解决，自然首先考虑相关的问题。 更简单的模型问题。 最近的研究表明，在适当的初始条件下，电子和带正电离子的无碰撞等离子体的双分量VlasovPoisson方程（2CVPE）的解具有许多性质，这些性质直接类似于具有涡面初始数据的不可压缩流体的二维涡度方程的解。 2CVPE是更简单的处理解析和更容易解决的数字。 这应该是一个非常有用的和重要的模式问题，建议的答案，开放的问题，涡面。 研究人员进行了数值研究的2CVPE和它的“粘性”正则化，两个组件的福克-普朗克方程。 我们的目标是确定这些方程组的解的性质时，初始数据是一个措施。 最近发现的2CVPE的精确解将集中这项工作。 研究人员还进行了数值研究的2-D不可压缩欧拉和Navier-Stokes方程与适当的涡面初始数据。 他利用从等离子体物理学相关模型问题的研究中获得的洞察力来激励和指导这项工作。 这项工作的目的是提供答案的许多开放性问题的解决方案的性质与涡面初始数据的欧拉方程。 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 物理学的基本定律，如质量守恒、动量守恒和能量守恒。 对于大多数问题来说，这个过程会导致一组太复杂而无法求解的数学方程。 基于物理或数学的直觉，科学家试图找到一个简化的公式的原始问题是足够简单的解决，但足够复杂的描述的基本特征的原始问题。 具有不同密度的流体层（如空气和水）或以不同速度移动的流体层之间的混合过程非常复杂，并且发生在各种重要应用中。 飞机上的气流也很复杂。 为了预测飞机后面的尾流，或者流体中的混合过程，科学家们经常使用一种被称为涡面的简化流体模型。 这不仅是对这些问题的最简单的现实描述，而且也是对许多其他大的牛顿数流的描述。 不幸的是，尽管涡面模型在流体动力学的许多问题中得到了广泛的应用，但对涡面的许多数学性质仍然知之甚少。 这个建议的目的是提供答案的涡面的数学性质的许多基本问题。 成功地完成拟议的工作将导致使用涡面模拟复杂的流体流动时的优势和局限性的理解增加。

9307726 Majda There are many unresolved fundamental problems about weak solutions of the incompressible Euler equations, such as vortex sheets, and regularizations of these weak solutions by smoothing the initial data or adding viscosity. Furthermore, since these problems are extremely challenging ones to try to solve with numerical methods, it is natural to first consider related simpler model problems. Recent research shows that the solutions of the two-component Vlasov Poisson equations (2CVPE) for a collisionless plasma of electrons and positively charged ions, with the initial condition being an appropriate measure, have many properties that are direct analogues of solutions of the 2-D vorticity equation for an incompressible fluid with vortex sheet initial data. The 2CVPE is simpler to treat analytically and easier to solve numerically. It should be an extremely useful and an important model problem for suggesting answers to the open problems about vortex sheets. The investigator conducts a numerical investigation of the 2CVPE and its "viscous" regularization, the two-component Fokker-Planc equation. The goal is to determine properties of solutions to these systems of equations when the initial data is a measure. Recently discovered exact solutions of the 2CVPE will focus this work. The investigator also conducts a numerical investigation of the 2-D incompressible Euler and Navier-Stokes equations with appropriate vortex sheet initial data. He uses the insight gained from the study of related model problems from plasma physics to motivate and guide this work. This work is designed to provide answers to many of the open problems about properties of solutions of the Euler equations with vortex sheet initial data. 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 probl ems this procedure leads to sets of mathematical equations that are too complicated to solve. Based on physical or mathematical intuition, the scientist then tries to find a simplified formulation of the original problem that is simple enough to solve, but sufficiently complicated to describe the essential features of the original problem. The mixing process between layers of fluid that 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. The flow of air over an airplane is also 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 the fluid called a vortex sheet. This is the simplest realistic description of not only these problems, but many other large Reynold's number flows. 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 fluid flows.