Systems of Nonlinear Partial Differential Equations in Transport Theory

9321383格拉斯西该奖项支持对非线性偏微分方程组的研究和对等离子体理论的应用。这项工作既涉及解析观点，也涉及数值观点。重点放在等离子体物理中出现的问题中的碰撞效应。这涉及到通过增加Boltzmann和Landau类型的碰撞算符来修改Vlasov-Maxwell系统。这项工作的主要目的是致力于研究朗道阻尼，其中规定了长时间行为的线性化的Vlasov方程的初值问题。人们已经在寻找相对论情况下的最优衰减率方面做了工作，在相对论情况下，由于忽略了碰撞效应，有效范围受到了限制。现在的重点将放在理解非相对论环境中的衰变。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。这部作品之所以重要，是因为它与宇宙审查制度有关。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

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. ***