On The Quantum Boltzmann Equation For Bose-einstein Particles

This thesis considers the quantum Boltzmann equation for Bose-Einstein parti-cles(BBE). For this equation, the3-D hard sphere and hard potential models are widelystudied and important results are obtained. In this thesis we shall study multi-D(N≥2)BBE with quantum collision kernels, including the2-D model. This model was firstintroduced by physicists Norheim, Uehling and Uhlenbeck in the early20th century.And its mathematical derivation had made important advancements in the21st century.The main results of the present thesis are as follows: Under certain assumptions on thecollision kernel, we prove the existence of global in time conservative measure solu-tions of homogeneous BBE for isotropic initial data. We also prove that, in the sense oflong-time behavior, there is a major diference between the measure solutions of2-Dand higher-D homogeneous BBEs. For the2-D case, the measure solution always con-verges strongly to the Bose-Einstein distribution as times goes to infinity, which meansthe Bose-Einstein condensation does not happen. For higher-D case, under the low-temperature condition, the measure solution converges strongly to the Bose-Einsteincondensation plus a Dirac-delta function. Here the presence of the Dirac-delta func-tion implies the occurrence of the Bose-Einstein condensation. These results coincidewith the common knowledge of the relation between the dimension and the occurrenceof Bose-Einstein condensation. When studying the long-times behavior of the measuresolution, the Bose-Einstein entropy plays an important role: We make sufciently useof the monotonicity of the entropy. And then, under the condition of relatively weakaveraged moment estimates, we prove the strong convergence of the measure solutionas t→+∞. The following are our future goals: Prove the present results with lessrestriction on the collision kernel; Study the possibility of finite-time condensation;Study the inhomogeneous BBE in bounded domains.The following are main innovations of the present thesis:The dimension of the velocity space is unrestricted. The collision kernels are chosen to be the quantum ones that have solid physicsbackground and cover well-known physical models for the case N=2.With relatively weak moment estimates, we prove the strong convergence toequilibrium for the measure solution.