Global well-posedness of a binary-ternary Boltzmann equation

Abstract

In this paper we prove global well-posedness for small initial data for the binaryternary 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 could possibly serve as a more accurate description model for denser gases in non-equilibrium. 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 solutions to appropriate linear problems. We show that the ternary operator allows consideration of softer potentials than the binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution.

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

“…(2. 14) We assume that K ě κ 0 admits the analytic properties of the pair correlations (2.77), (2.88) below. The constant Z 0 :" Trpe ´Kq (2.15) is chosen to obtain the normalization ρ 0 p1q " 1.…”

“…The well-posedness problem for the Boltzmann equation is a large and important research area; we will not address questions of this type in this work, but refer, for instance, to [4,5,6,7,8,9,10,14,19,34,35,25,50,52,53,54,78,81,85,86,105,107,110,108,116,117,149,155,156,157,158,165,202,201,203,207,208,209,211] and references therein.…”