Global well-posedness of a binary-ternary Boltzmann equation

TL;DR

This paper proves the global well-posedness of a binary-ternary Boltzmann equation, extending classical results to include ternary interactions and providing a more accurate model for denser gases.

Contribution

It introduces a new analysis for the binary-ternary Boltzmann equation, establishing global well-posedness and novel convolution estimates for ternary operators.

Findings

Established global well-posedness near vacuum for the binary-ternary Boltzmann equation.

Developed new convolution estimates for ternary collisional operators.

Showed ternary operators allow softer potentials while preserving properties of binary interactions.

Abstract

In this paper we show global well-posedness near vacuum for the binary-ternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and may serve as a more accurate description model for denser gases in non-equilibrium. Well-posedness of the classical Boltzmann equation and, independently, the purely ternary Boltzmann equation follow as special cases. To prove global well-posedness, we use a Kaniel-Shinbrot iteration and related work to approximate the solution of the nonlinear equation by monotone sequences of supersolutions and subsolutions. This analysis required establishing new convolution type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the one binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution. These results are novel for collisional operators of monoatomic gases with either hard or soft potentials that model both binary and ternary interactions.