Contribution to the mathematical study of the Boltzmann and Landau equations in kinetic theory of gases and plasmas.

Summary We study kinetic equations of the form f/t + v. *#xf = q(f,f), t0, x ,r#n, v , r#n, which describe the evolution of a gas or a plasma in which the particles undergo collisions modeled by the operator q, known as Boltzmann's -: q(f,f) dv#* d b(v v#*, ) (f'f'#* ff#*), - or Landau: q(f,f) = /v#i dv#*a#i#j(v v#*) (f#*f/v#j ff#*/v#*#,#j). The first three parts of this thesis concern the homogeneous (x-independent) equations #tf = q(f,f). Two points are particularly emphasized: grazing collisions and entropy dissipation. In the first part, we study the Cauchy problem associated to the homogeneous Boltzmann and Landau equations (for realistic and possibly singular cross sections), the regularity properties of the solutions, as well as the asymptotics of grazing collisions which allows us to pass from one equation to the other. The second part is devoted to the special case of Maxwellian molecules. Under this assumption, a detailed study is made of the Cauchy problem and of the return to equilibrium, as well as of the links between kinetic theory and information theory. In the third part, we use the previous results to obtain explicit estimates of the rate of return to equilibrium in the general case. In the fourth part, we obtain partial results on the Cauchy problem for the inhomogeneous Landau equation, and some new estimates on the inhomogeneous Boltzmann equation. Finally, in the fifth part, we establish various conservative forms of the Boltzmann operator, with application to the asymptotics of grazing collisions.