Asymptotic Limit of Solutions
of the Boltzmann Equation
In this lecture, we present recent results about the large time behavior of kinetic equations with collision kernels (in particular results on the Boltzmann and Landau equations appeared in papers by G. Toscani, C. Villani and myself).
We first present estimates relating the entropy dissipation and the entropy for various collision kernels. Then, we show how these estimates can be used to get explicit rates of convergence in the inhomogeneous setting (in the whole space with a confining potential, or in a bounded box with specular reflexion).
Among the models that we wish to study, we quote the (linear) Fokker-Planck equation, the transport equations of neutrons and photons (radiative transfer), the Landau equation of plasma physics, the Boltzmann equation of rarefied gases, the Boltzmann equation for semiconductors, the BGK model.
We compare systematically the methods based on the entropy with the methods based on spectral theory and linearization.