Existence of solutions to the hard sphere Boltzmann equation

We will prove the global existence and uniqueness of solutions to the Cauchy problem for the Boltzmann equation of a hard sphere, assuming small data. We will also show that such solutions, assuming nonnegative initial data, remain nonnegative, which is what we expect, as a solution represents a density. We follow chapter 2 of The Cauchy Problem in Kinetic Theory by Glassey.