The Relativistic Quantum Boltzmann Equation Near Equilibrium

The relativistic quantum Boltzmann equation (or the relativistic Uehling-Uhlenbeck equation) describes the dynamics of single-species fast-moving quantum particles. With the recent development of relativistic quantum mechanics, the relativistic quantum Boltzmann equation has been widely used in physics and engineering, for example in the quantum collision experiments and the simulations of electrons in graphene. In spite of such importance, there has, to the best of our knowledge, been no mathematical theory on the existence of solutions to the relativistic quantum Boltzmann equation. In this paper, we prove the global existence of a unique classical solution to the relativistic Boltzmann equation for both bosons and fermions, when the initial distribution is nearby a global equilibrium.