Global Existence, Uniqueness And Long Time Behavior Of Solutions Of A Quantum Boltzmann Equation

The Boltzmann equation is an importance model in statistics physics and kinetic the-ory, which descries the time evolution of the dilute gas. When we are concerned with thequantum efect of particles, the classical Boltzmann equation give an inadequate descrip-tion of the kinetic evolution of the system, we have to consider the quantum Boltzmannequation. Such equation has been used in the condensation of the low temperature gasand the design of the semiconductor and so on.In this paper we consider the dilute system of identical bosons (or fermions) con-fined in an isotropic harmonic oscillator. The time evolution of the density distribution ofthe system can be described by a semi-discrete quantum Boltzmann equation. This modelcomes from physics literatures, which is derived from the Schro¨dinger equation for themany body system in quantum theory. Due to the quantum mechanics, the distributionfunction, i.e. the solution, is discrete in the energy variable. Although this quantumBoltzmann equation has been derived by physicians, so far there is no mathematical re-sults about existence, stability and long time behavior of solutions to the equation. Inthe present paper we give the classification of equilibria of the equation for bosons andfermions respectively, and prove the global existence, uniqueness and the strong conver-gence to equilibrium for solutions of the equation. The main results are the followingthree parts:Firstly, by solving the equilibrium equations and the moment system, we obtain theclassification of the equilibrium for bosons and fermions respectively. Since the equilib-rium is a discrete sequence for this semi-discrete model, the equilibrium equation and themoment system are diferent from the ones for the continuous model. In particular, thesolvability of the moment system becomes more difcult. Therefore we make use of theglobal inverse function theorem to overcome this difculty.Secondly, we introduce a weight l1space as the solution space, and using the con-traction mapping theorem and the conversation of mass and energy to prove the globalexistence and uniqueness of solutions. Then by constructing the positive functions ap-proximating to solutions and making use of the convergence argument, we obtain theentropy identity. Finally, we give some entropy inequalities and calculate the moment estimates ofsolutions. Assuming the high order moment of the initial data is bounded, we showthat although the high order moment of the solution is increasing with respect to time,but the mean value of the moment is uniformly bounded. With the help of the entropyinequalities and the moment estimates, and using the compactness of the solution space,we prove strong convergence to equilibrium of the solution.