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.
