High frequency limit for Klein-Gordon-Maxwell equations

Overview & Synopsis

The Klein-Gordon-Maxwell equations are used in several physical contexts, as a simple model for gauge theories,in quantum electrodynamics... As for other hyperbolic or elliptic equations, it is interesting to study the behaviorof high-frequency solutions. Especially at the limit, when the wavelength tends to 0.In the case of the vacuum Einstein equations, the Burnett conjecture states that a high-frequency limit of solutions (in a sense to be specified)is a solution to the massless Einstein-Vlasov equations. This phenomenon of backreaction is linked to the nonlinearities of the Einstein equations.For KGM, the nonlinearities are weaker and have a more flexible structure, which suggests the existence of more singular high-frequency solutions.Moreover, the presence of the Planck constant h (representative of the quantum effects in KGM) in the equationsallows the study of another type of high-frequency solutions, the ones appearing at the semi-classical limit,that is, when h tends to 0.The aim of this thesis will be to identify possible limits (semi-classical or not) for sequences of high-frequency solutionsto the Klein-Gordon-Maxwell equations and to understand which relativistic fluid or kinetic system are they solution to. To do so, several methods will be used: geometric optics,semi-classical analysis, modulated energy method, study of simplified equation (nonlinear Klein-Gordon)...