Global existence and large time behavior of the quantum Boltzmann equation with small relative entropy

In this paper, we study the global well-posedness of the non-relativistic quantum Boltzmann equation with initial data of small relative entropy. For a class of initial data which are allowed to have arbitrary bounded amplitude and even contain vacuum, we establish the global existence and uniqueness of the mild solutions to the quantum Boltzmann equation in the torus \begin{document}$ x\in\mathbb T^3 $\end{document}. The exponential time decay rate is also obtained in the \begin{document}$ L^{\infty}_{x, v} $\end{document}-norm.