Numerical Investigations of the Incompressible Euler Equations, the Vlasov-Poisson/Fokker-Planck Equations, and the Boltzmann Equation

摘要：本文提出用数值方法解决具有光滑初始数据的三维不可压缩欧拉方程的有限时间奇点发展问题。这些方程模拟了不可压缩理想流体的流动。特别是，我将考虑轴对称与漩涡的情况下，允许更高的分辨率比可能在全三维实验。将实现几种不同的有限差分技术。我将研究不同数值方案的性能，以评估每种方法的质量。此外，我将研究不同数值边界条件对涡度放大的影响，以及使用粒子方法的全三维问题。结合这些实验，我将在一维上对Vlasov-Poisson方程进行数值模拟。这些方程模拟了均匀离子背景下的无碰撞电子等离子体，并作为二维不可压缩欧拉方程的简单模拟。我将数值研究由非光滑电子薄片初始数据引起的Vlasov-Poisson和Fokker-Planck-Poisson方程弱解的行为。电子薄片描述了一束集中的电子束。这些方程将通过平滑初始条件或包括由福克-普朗克-泊松方程模拟的碰撞来正则化。我建议使用杰克·谢弗（Jack Schaeffer）的有限差分法和粒子法来检验在消失正则化极限下得到的Vlasov-Poisson方程的解。最后，这个提议的研究将为我以后进行与具有潜在医学应用的非月平滑初始数据的玻尔兹曼和福克-普朗克方程相关的研究做准备。这些方程用于放射治疗中剂量计算的计算。

Abstract: I propose to numerically address the question of finite time singularity development in the three-dimensional incompressible Euler equations with smooth initial data. These equations model the flow of incompressible ideal fluids. In particular, I will consider the axisymmetric with swirl case which allows for higher resolution than possible in fully three-dimensional experiments. Several different finite difference techniques will be implemented. I will investigate the performance of the different numerical schemes in order to assess the quality of each method. Also, I will study the effects different numerical boundary conditions have on vorticity amplification as well as the fully three-dimensional problem using particle methods. In conjunction with these experiments, I will perform numerical simulations of the Vlasov-Poisson equations in one dimension. These equations model a collisionless plasma of electrons in a uniform background of ions, and serve as a simpler analogue of the two-dimensional incompressible Euler equations. I will numerically study the behavior of weak solutions to the Vlasov-Poisson and Fokker-Planck-Poisson equations arising from non-smooth electron sheet initial data. An electron sheet describes a concentrated beam of electrons. The equations will be regularized by either smoothing the initial condition or by including collisions modeled by the Fokker-Planck-Poisson equations. I propose to use both a a finite difference method developed by Jack Schaeffer as well as particle methods to examine the solution of the Vlasov-Poisson equations obtained in the limit of vanishing regularization. And finally, this proposed research will prepare me to afterward conduct research related to the Boltzmann and Fokker-Planck equations with non-qmooth initial data which has potential medical appliations. These equations are used in the computation of dosage calculations in radiation therapy.