Global well-posedness of a binary–ternary Boltzmann equation

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 non-linear 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, and 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.