The Boltzmann equation and corresponding extremal problems

TL;DR

This paper introduces a new distance measure between solutions of the Boltzmann equation and a global Maxwellian, and solves an extremal problem to find solutions minimizing this distance under fixed energy and moments.

Contribution

It generalizes the Kullback-Leibler distance for Boltzmann solutions and solves an extremal problem for minimal distance with fixed energy and moments.

Findings

Defined a new distance measure between Boltzmann solutions and Maxwellian.

Solved the extremal problem for minimal distance with fixed energy and moments.

Extended the concept of Kullback-Leibler distance in kinetic theory.

Abstract

We start with some global Maxwellian function $M$, which is a stationary solution (with the constant total density $\rho$) of the Boltzmann equation, and we denote the number of the corresponding space variables by $n$. The notion of distance between the global Maxwellian function and an arbitrary solution $f$ (with the same total density $\rho$ at the fixed moment $t$) of the Boltzmann equation is introduced. In this way we essentially generalize the important Kullback-Leibler distance, which was used before. An extremal problem to find a solution of the Boltzmann equation, such that $\dist\{M,f\}$ is minimal in the class of solutions with the fixed values of energy and of $n$ moments, is solved.