Analysis of kinetic and diffusive multi-species systems

Abstract

The objective of this thesis is the analysis of kinetic and diffusive multi-species systems with certain cross effects between the species, which are very important in many applications in physics, biology and chemistry. In the first part of this thesis, we study the multi-species Boltzmann equation for hard potentials or Maxwellian molecules under Grad's angular cut-off condition on the torus, which describes the evolution of a dilute gaseous mixture. First, we work on the linearized level with same molar masses, where we prove a multi-species spectral-gap estimate of the collision operator, which leads to exponential trend to global equilibrium using the hypocoercive properties of the linearized Boltzmann equation. Next, we study the full Cauchy theory of the nonlinear multi-species Boltzmann equation close to global equilibrium for different molar masses in physically relevant function spaces, recovering the optimal physical threshold in the particular case of a multi-species hard spheres mixture with same molar masses. The second part of this thesis is devoted to cross-diffusion systems. First, we prove global existence of weak solutions for a generalized SKT cross-diffusion population dynamics model with an arbitrary number of species under detailed balance or weak cross-diffusion condition using entropy methods. Finally, we study a rigorous fast-reaction limit from reaction-diffusion systems to cross-diffusion systems using entropy estimates and additional duality estimates. Since the reaction-diffusion system exhibits an entropy structure, performing the fast-reaction limit leads to a limiting entropy of the limiting cross-diffusion system. In this way, we obtain new entropies for new classes of cross-diffusion systems.