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

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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.