Analysis of models of mathematical physics.

Summary The works gathered in this thesis are related to the mathematical treatment of two distinct topics in mathematical physics. In part A: minimization problems in hadronic matter physics, we find results concerning variational models used in nuclear physics to describe the strong interaction in the low energy limit. The existence of solutions realizing the energy infimum with a constraint of topological degree for different models of the Skyrme type is discussed, with emphasis on the choice of appropriate functional spaces. We use the concentration-compactness method (because of the compactness defects due to the translation invariance), and different functional analysis techniques. We then study a model with non-local coupling, the Adkins and Nappi model, in the framework of the Skyrme ansatz, using in particular the local behavior of the solutions of the Euler-Lagrange equations. In part B: kinetic equations several models describing rarefied gases or plasmas are discussed. We start by studying a modified Boltzmann equation to take into account the quantum corrections for a fermion gas (existence, uniqueness, conserved quantities, h theorem, convergence to the classical limit to the ordinary Boltzmann equation, asymptotic solutions in large time and stationary solutions). We then study the stationary Maxwellian solutions of the Vlasov-Poisson system representing charged particles subjected to the mean field created by their charge distribution and to an external electrostatic field ensuring their confinement (existence and uniqueness in Marcinkiewicz spaces). The last chapter deals with the convergence to asymptotic states in large time for the Vlasov-Poisson-Boltzmann system and the study of the corresponding stationary solutions in the case where the mass and energy are conserved.