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

TL;DR

This paper proves the existence of global regular solution branches for the Boltzmann equation in higher dimensions and identifies multiple solution types, including weakly singular solutions, highlighting the equation's complex solution structure.

Contribution

It establishes the existence of global regular solutions with polynomial decay and demonstrates the presence of weakly singular solutions in higher dimensions.

Findings

Existence of global regular solution branches in phase space dimension 2d≥6.

Presence of weakly singular solutions in spatial dimension d≥3.

Solutions can have infinite relative entropy with respect to Gaussian.

Abstract

Existence of global regular solution branches of the Boltzmann Cauchy problem with continuously differentiable data in phase space dimension $2d\geq 6$ 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 $d\geq 3$, 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.