Rate Of Convergence To Equilibrium For Some Collisional Kinetic Models

In this thesis,we consider the exponential convergence rate to equilibrium for some collisional kinetic models,including the linear Boltzmann equation for hard potentials without angular cutoff,the fermion equation and the ellipsoidal BGK model.All the above models are analyzed from a spectral point of view and from the point of view of semigroups.Our strategy is taking advantage of a spectral gap estimate in smaller reference Hilbert space,the factorization method and the enlargement of the functional space developed by Gualaani.Mischler,and Mouhot.In Chapter 1,we first give a brief introduction to the background and known results of the researches in this thesis.Then,we state our main results and give some comments on the main results.Finally,we give some notations,definitions and some functional results.In Chapter 2.we consider the asymptotic behavior of solutions to the linear Boltzmann equation for hard potentials without angular cutoff:(?)We first deal with the linear spatially homogeneous Boltzmann equation and ob-tain an optimal rate of exponential convergence towards equilibrium in L1-space with a polynomial weight.Then we investigate the linear spatially inhomoge-neous Boltzmann equation and obtain the exponential decay in Lv1Lx2-space with a polynomial weight.In Chapter 3,we study the asymptotic behavior of solution to the fermion equation in the Sobolev spaces with a polynomial weight in the torus:(?)We first investigate the linearized equation and obtain the optimal exponential decay rate for the associated semigroup.Our strategy is taking advantage of quantitative spectral gap estimates in smaller reference Hilbert space,the factor-ization method and the enlargement of the functional space.We then turn to the nonlinear equation and prove the global existence and uniqueness of solutions in a close-to-equilibrium regime.Moreover,we obtain an exponential stability for such a solution with the optimal decay rate given by the semigroup decay of the linearized equation.In Chapter 4.we deal with the asymptotic behavior of solution to the lin-earized ellipsoidal BGK model in the torus:(?)We prove that the solution converge towards the equilibrium in the weighted Sobolev spaces with a polynomial weight.We establish the optimal exponential decay rate for the associated semigroup.Our strategy is taking advantage of quantitative spectral gap estimates in smaller reference Hilbert space,the factor-ization method and the enlargement of the functional space.