Global Existence And Long Time Behavior Of Anisotropic Solutions Of The Boltzmann Equation For Bosons

In this thesis we investigate two problems:1.The global in time existence and uniqueness of solutions of the spatially homo-geneous Boltzmann equation for Bose-Einstein particles for anisotropic initial data.2.The convergence to equilibrium and the rate of convergence of the solutions that we have obtained.The quantum Boltzmann equation for Bose-Einstein particles?BBE?describes the time evolution of dilute Bose gases.Because of quantum effects,the BBE equation in-cludes cubic terms of solutions which are more complicated than the quadratic terms of solutions in the classical Boltzmann equation and they make it difficult to apply main methods in the study of classical Boltzmann equation.So far,even for spatial-ly homogeneous BBE equation,results of global in time existences of solutions and other physic effects are concerned with isotropic initial data?which implies that the solutions are also isotropic?.As for anisotropic initial data,the first result was that Briant and Einav^[1]proved local in time existences and uniqueness of solutions with isotropic initial data of the BBE equation in 2016.This thesis includes the latest devel-opment concerning anisotropic initial in which we prove the global in time existence,uniqueness and long time behavior of solutions of the spatially homogeneous BBE e-quation for the hard sphere model for bounded anisotropic initial data with necessary high temperature assumption.Our strategy to prove the global existence of anisotropic solutions is:Consider an intermediate equation with the truncated cubic terms.Thus the intermediate e-quation's main properties are similar to the classical Boltzmann equation's properties which makes it relatively easy to prove the global existence and uniqueness of the so-lution f of the intermediate equation.Then we use the multi-step iterations of the collision gain operator to obtain a desired uniform L^?-bound of f.Finally,we prove that if the initial data are small relative to planck constant the truncation in the inter-mediate equation can be removed so that f becomes the global solution of the BBE equation.Our main idea of the proof of the strong convergence to the Bose-Einstein distri-bution of the solution f is that it could be converted to prove the strong convergence of f/?1+f?to a suitable Maxwell distribution.Finally,when we estimate the rate of convergence to equilibrium,to use the Villani's entropy inequality,we use a convex combination of f which makes the logarithm term only has the low term increasing to establish a differential inequality to estimate the decay rate.Solving the inequality and making further estimate we can prove the algebraic rate of convergence of the solution f.