On the Convergence of Discrete Kinetic Approximations to Hydrodynamic Equations

Introduction Motivations This thesis is devoted to present some new results in the theory of the semilinear approximations to hydrodynamic equations. More specically, the main purpose of this investigation is to prove the convergence of a class of discrete kinetic approximations , the BGK models, to some hyperbolic and hyperbolic-parabolic systems. These BGK models were introduced by Bathnagar, Gross, and Krook as a continuous velocities simplication of the Boltzmann equation, and during the last twenty years they have been generalized, both with continuous and discrete velocities, in order to construct kinetic equations associated with dierent hydrodynamical systems (see for instance [41], which is a textbook about continuous velocities approximations). In this work we consider the following class of semilinear system of balance laws with source with a nite set of constant velocities: