Clément Mouhot; Cédric Villani; École normale supérieure (Lyon).
|描述：||1 vol. (461 p.) ; 30 cm.|
|責任：||Clément Mouhot ; sous la direction de Cédric Villani.|
We are interested in this PhD in the study of solutions to the Boltzmann equation (elastic or inelastic) and the Landau equation. The axis of this study are the regularity and asymptotic behavior of the solutions, and we systematically search for quantitative results. In the first part, we consider on the one hand the spatially homogeneous solutions to the Boltzmann equation, for which we prove propagation of regularity and damping of singularities for short-range interactions, as well as propagation of integrability bounds for long-range interactions. On the other hand, we quantify the positivity of the spatially inhomogeneous solutions, under regularity assumptions. In the second part, we give spectral gap and coercivity estimates for the linearized Boltzmann and Landau operators, and we prove exponential convergence to equilibrium with explicit rate for a gas of spatially homogeneous hard spheres. In the third part, we consider the spatially homogeneous Boltzmann equation for granular gases, for which we construct solutions for realistic models of inelasticity (however strongly non-linear) and discuss the possibility of cooling in finite time or asymptotically. We then show the existence of self-similar profils, and study the behavior of solutions for large velocities. In the forth part, we use a semi-discretization of the Boltzmann operator in order to propose fast numerical schemes based on the spectral method or discrete velocity models.