Laser-Matter Interactions and Highly Nonlinear Geometrical Optics; Dynamics of Reacting Flows

摘要：本项目研究描述a)激光-物质相互作用和b)反应流的模型方程的数学性质，特别是：a)Maxwell-Euler和Maxwell-Landau方程的高频、大振幅解。建立在物理基本方程基础上的方程组太复杂了，不能作为数值模拟的基础。因此，需要简单的模型系统。这个项目解决了描述激光-物质相互作用的模型系统的有效性问题，例如Zakharov和Davey-Stewartson模型。B)反应流的稳定性问题。该项目将通过研究简单模型系统的分叉，为反应流中发生的一维不稳定性提供简单的数学描述。最终，将对反应的Navier-Stokes方程的更复杂的框架进行分析，其中将不得不使用使用逐点格林函数界限的最新技术。这些项目的动机来自实际实验：高能激光的大规模实验显示出重要的能量损失；爆轰波被认为会产生纵向不稳定性。该项目将有助于对描述这些现象的相关模型方程进行严格的数学分析。这种分析是开发预测工具的关键一步，因为需要对模型有深刻的数学理解，以便设计出有效的数值模拟。

Abstract: This project investigates mathematical properties of models equations describing a) laser-matter interactions, and b) reacting flows, specifically: a) High-frequency, large-amplitude solutions of the Maxwell-Euler and Maxwell-Landau equations. Systems of equations based on the fundamental equations of physics are too complex to serve as a basis for numerical simulations. Hence the need of simple model systems. This project addresses the question of the validity of model systems describing laser-matter interactions, such as the Zakharov and the Davey-Stewartson models. b) Stability issues for reacting flows. The project will provide a simple mathematical description of one-dimensional instabilities occurring in reacting flows by studying bifurcations of simple model systems. Ultimately, the analysis will be carried out to the more complex framework of the reacting Navier-Stokes equations, where recent techniques using pointwise Green's functions bounds will have to be used. The motivation for these projects comes from actual experiments: large-scale experiments of high-energy lasers show important enerngy losses; detonation waves are seen to develop longitudinal instabilities. This project will contribute to a rigorous mathematical analysis of relevant model equations describing these phenomena. Such an analysis is a key step in the development of predictive tools, as a deep mathematical understanding of the models is needed in order to devise efficient numerical simulations.