Systems of Nonlinear Partial Differential Equations in Transport Theory

9321383 Glassey该奖项支持对非线性偏微分方程组的研究以及在等离子体理论中的应用。这项工作涉及分析和数值两方面的观点。重点是等离子体物理问题中的碰撞效应。这涉及到通过增加玻尔兹曼型和朗道型的碰撞算符来修改弗拉索夫-麦克斯韦方程组。本文的主要工作是研究朗道阻尼，其中规定了线性化Vlasov方程的初值问题的长时间行为。在相对论的情况下，人们已经在寻找最优衰变率方面做了一些工作，在相对论的情况下，有效的范围受到忽略碰撞效应这一事实的限制。现在重点将放在理解非相对论背景下的衰变。还将对弗拉索夫-爱因斯坦系统进行研究，以包括恒星动力学问题的完全相对论模型。这项工作很重要，因为它与宇宙审查制度有关。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

9321383 Glassey This award supports research on systems of nonlinear partial differential equations and applications to plasma theory. The work involves both analytical and numerical points of view. The emphasis is on collisional effects in problems arising in plasma physics. This involves the modification of the Vlasov- Maxwell systems through the addition of collision operators of Boltzmann- and Landau-type. The major thrust of this effort is dedicated to the study of Landau damping, in which the long time behavior the initial value problem for the linearized Vlasov equation is prescribed. Work has already been done on finding optimal decay rates in the relativistic case, in which the range of validity is restricted by the fact that collisional effects are neglected. Emphasis will now be placed on understanding decay in the nonrelativistic setting. Studies will also be made on the Vlasov-Einstein system to include a fully relativistic model for stellar dynamic problems. The work is important because of its relation to cosmic censorship. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***