Global existence and stability of solutions of spatially homogeneous Boltzmann equation for Fermi-Dirac particles

Abstract

This paper deals with the spatially homogeneous Boltzmann equation for Fermi-Dirac particles for hard sphere model. Firstly, we prove the global existence and uniqueness of classical solutions to this problem and give the corresponding L∞ and L31 estimates of solutions. To achieve this aim, we give the global existence of an intermediate equation which behaves as classical Boltzmann equation, then we prove that the intermediate solutions become the original solutions when the L∞-bound of intermediate solutions less than a fixed constant. Then using the uniformly bounded L31 estimation, we prove the stability of spatially homogeneous Boltzmann equation for Fermi-Dirac particles. In addition, an upper bound of L31 estimate of classical Boltzmann equation at infinite time interval is given. Using this useful estimate we prove the stability between quantum Boltzmann equation and classical Boltzmann equation when the Planck constant tends to zero.