Monotonicity methods in non-linear kinetic theory

The last years have been marked by an increased interest in the mathematical properties of the non-linear kinetic models, appearing as generalizations of the classical Boltzmann equation. This can be explained by various applications not only in physics, astrophysics and chemistry (e.g. study of simple and complex/reacting fluids, granular media, coagulation-fragmentation, formation of planetary rings, galaxy collision) but also in modeling evolution processes in immunology, traffic flow, communication networks, etc. The above generalized Boltzmann equations are proved to present some mathematical properties similar to that of the classical Boltzmann equation. Due to this fact, the problem of the existence, uniqueness and positivity of global solutions can be solved by extending, non-trivially, monotonicity methods developed within the framework of the mathematical kinetic theory of the classical Boltzmann equation. (author)