Well-posedness and Behavior of Solutions to Kinetic Equations

点击翻译按钮获取中文摘要

This project will develop new analytic and computational methods to solve a variety of mathematical problems in the kinetic theory of plasma dynamics. Plasmas are often referred to as the fourth state of matter (after solids, liquids, and gases) and account for 99.99% of all material in the universe. Since plasmas are charged gases, they serve as excellent conductors of electricity, and thus are of great practical interest. As an example, plasma engines have been developed by a number of space agencies and recently used to power some NASA spacecraft. Additionally, the use of plasmas within nuclear fusion as a source of clean energy is currently of immense scientific interest. Other notable examples of plasmas include the solar wind, the Earth's ionosphere, galactic nebulae, and comet tails. A complete understanding of the solar wind would also be particularly useful, as this natural phenomenon dictates the intensity of "space weather", which is often responsible for expensive damage to satellites orbiting the Earth. In general, the motion of a plasma is described by a number of complicated partial differential equations dictated by physics. Among the goals of the current project are to demonstrate that these equations possess realistic solutions, determine their qualitative behavior, compute their sensitivity with respect to model parameters (such as masses, charges, and temperature), and computationally approximate them so that one can accurately predict outcomes in future situations.A plasma is a fully ionized gas in which electromagnetic forces are often strong enough to dominate collisional effects. The motion of a high temperature, low density collisionless plasma is described by the Vlasov-Maxwell equations, a nonlinear system of hyperbolic partial differential equations. In this setting, collisions are neglected while the charge and current densities, which drive the Maxwell system, are determined in a self-consistent manner from velocity averages of solutions to the Vlasov equation. One major question this project will study is: are there shocks in a collisionless plasma? That is, could a singularity develop from smoothly prescribed initial values as time progresses? In some cases, such as in lower dimensional, relativistic formulations (e.g., one space and two velocity variables), smooth global solutions are known to exist. Similarly, the associated Fokker-Planck system will be studied to further elucidate the smoothing influence of collisions on the particle distribution. Another problem to be investigated concerns the long-time behavior of the charge density and electromagnetic field in the system. More specifically, do dispersive effects in the equations cause these quantities to decay over time, or is there sufficient interaction so as to sustain their strength even as time tends to infinity? Finally, the sensitivity of fields and densities with respect to model input parameters will be computed using global sensitivity metrics and particle-in-cell simulations.