The Binary-ternary Boltzmann Equation Near Equilibrium

The classical Boltzmann equation, formulated by Boltzmann in 1872, describes the statistical behavior of a dilute gas in a non-equilibrium state. While the global well-posedness of the classical Boltzmann equation has been extensively studied and is relatively well-understood, this model becomes inadequate when the gas is no longer sufficiently dilute. Therefore, it is necessary to investigate higher-order extensions of the Boltzmann equation that account for multi-particle interactions. Recently, Ampatzoglou and Pavlović provided the rigorous derivation of both the ternary [$\mathit{\text{Comm.}}\mathit{\text{Math.}}\mathit{\text{Phys.}}$ $\mathbf{\text{387}}$ (2021), no. 2, 793-863.] and the binary-ternary [$\mathit{\text{Forum}}\mathit{\text{Math.}}\mathit{\text{Sigma}}$ $\mathbf{\text{13}}$ (2025), Paper No. e52, 95 pp.] Boltzmann equations from a classical particle system, yielding an explicit form of the ternary collision operator. However, the global existence near equilibrium for the binary-ternary Boltzmann equation remains an interesting and fundamental question, a gap that this paper seeks to address as its primary contribution.