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

9303414，PI-Zheng：郑玉熙提出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。 他对等离子体初始状态非常不规则的情况感兴趣。即电子和离子的密度是有限的。因此，一个主要的困难是找到正确的弱公式，在其中可以证明存在一个并且可能是唯一的弱解。 当使用衡量价值解决方案的概念时，他会仔细寻找证据来证明这是不可避免的。 如果弱解不唯一或不稳定，他将寻找适当的熵准则和稳定性条件或不稳定的潜在机制。 所使用的方法包括Vol'pert的$BV$微积分、DiPerna-Majda的浓度对消法以及Calder\'on-Zygmund型的势能理论。 明确的精确解决方案是可能的，他将广泛构建和使用它们。 他还将研究弱解的数值格式的弱收敛性。 郑玉熙提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感兴趣。该问题在数学上类似于具有奇异初始数据的不可压缩流体流的运动。 这两个问题都具有物理意义并提供数学挑战。 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。 了解这些不规则运动将有助于非线性现象的整个前沿，并且无疑会改进这些领域的现有技术。

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.