Mathematical Sciences: 1-D Vlasov-Poisson and 2-D Euler Equations with Measures as Initial Data

数学科学：以测量值作为初始数据的一维弗拉索夫-泊松方程和二维欧拉方程

• 批准号: 9303414
• 负责人: Yuxi Zheng
• 金额: $ 6.03万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-12-15 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303414&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Vlasov; Poisson; Euler

9303414, PI-Zheng: Yuxi Zheng proposes to study the one-dimensional Vlasov-Poisson equations describing the motion of a collisionless plasma (or stellar dynamics). He is interested in cases when the initial state of the plasma is very irregular; i.e., the densities of electrons and ions are finite measures. Thus, a major difficulty is to find the right weak formulation in which one can prove that there exists one and possibly the unique weak solution. When the concept of measure-valued solutions is used, he will carefully look for evidence to show that it is inevitable. If weak solutions are not unique or stable, he will look for appropriate entropy criteria and stability conditions or the underlying mechanism of instability. The methods to be used include the $BV$ calculus of Vol'pert, the concentration-cancellation method of DiPerna-Majda, and potential theory of Calder\'on-Zygmund type. Explicit exact solutions are possible and he will construct and use them extensively. He will also investigate weak convergence of numerical schemes for weak solutions. Yuxi Zheng proposes to study the motion of a collisionless plasma. He is interested in cases when the initial state of the plasma is very irregular. This problem resembles mathematically the motion for an incompressible fluid flow with singular initial data. Both problems are of physical significance and provide mathematical challenge. Physical examples in clude irregular motions in galaxy formation, controlled nuclear reactors, and water or air wakes of fast moving bodies. Understanding of these irregular motions will help the entire front of nonlinear phenomena and undoubtedly improve the current technologies in these areas.

9303414，郑玉玺：郑玉玺建议研究描述无碰撞等离子体运动(或恒星动力学)的一维弗拉索夫-泊松方程。他对等离子体的初始状态非常不规则的情况很感兴趣，即电子和离子的密度是有限的测量。因此，一个主要的困难是找到正确的弱公式，在这个公式中人们可以证明存在唯一的弱解，并且可能存在唯一的弱解。当使用度量值解的概念时，他会仔细寻找证据来证明这是不可避免的。如果弱解不是唯一的或不稳定的，他将寻找合适的熵标准和稳定性条件或不稳定的潜在机制。所用的方法包括Vol‘pert的$BV$演算、DiPerna-Majda的浓度抵消方法和Calder’on-Zygmund型势理论。显式精确解是可能的，他将广泛地构造和使用它们。他还将研究弱解的数值格式的弱收敛。郑玉玺建议研究无碰撞等离子体的运动。他对等离子体初始状态非常不规则的情况很感兴趣。这个问题在数学上类似于具有奇异初始数据的不可压缩流体的运动。这两个问题都具有物理意义，并提供了数学挑战。物理例子包括星系形成中的不规则运动，受控核反应堆，以及快速移动物体的水或空气尾迹。对这些不规则运动的了解将有助于整个非线性现象的前沿，并无疑将改进这些领域的现有技术。