On global regular solution branches and multiple solutions of the Boltzmann equation

Existence of global regular solution branches of the Boltzmann Cauchy problem with continuously differentiable data in phase space dimension $2d\ge 6$ with polynomial decay at infinity of order greater than $2d$ is proved. There are data in this class of infinite relative entropy with respect to the Gaussian. Furthermore, there are weakly singular solution branches of the Boltzmann equation in spatial dimension $d\ge 3$, i.e., solutions of the Boltzmann equations which are only Lipschitz with respect to the velocity variables at some point in phase space. This is in accordance with a.e. $L^1$-uniqueness of renormalized solutions (cf.\cite{L}) and more classical results in function spaces of mixed regularity.