Classical solutions to the Boltzmann equations for gas mixture with unequal molecular masses

TL;DR

This paper establishes the global existence of classical solutions to the Boltzmann equation for gas mixtures with unequal molecular masses, broadening understanding beyond single-species models and aiding future spectral analysis.

Contribution

It provides the first detailed analysis of the linear collision operator and nonlinear estimates for mixtures with unequal molecular masses, extending existing theories.

Findings

Proves global in time existence of classical solutions near Maxwellians.

Characterizes the structure of the linear collision operator for unequal masses.

Establishes estimates for nonlinear terms under soft potential conditions.

Abstract

The Boltzmann equation is essential for gas thermodynamics,as it models how the molecular density distribution $F(t,x,v)$ changes over time. However, existing research primarily focuses on the single species Boltzmann equation, while investigations into gas mixtures with unequal molecular masses remain relatively limited. Notably, mixed gas studies have broader applications exemplified by Earth's atmosphere, composed of 78\% nitrogen, 21\% oxygen, and 1\% trace gases, where the $N_2$ to $O_2$ molecular mass ratio is 28:32 (simplified as 7:8). This work addresses the Boltzmann equations for such mixtures with unequal molecular masses $(m^A\neq m^B)$, establishing the global in time existence of classical solutions near Maxwellians for soft potentials ($-3<\gamma<0$) in a periodic spatial domain. Our analysis encompasses arbitrary molecular mass ratios. Our analysis encompasses arbitrary molecular mass ratios. The main contribution of this paper lies in the detailed characterization of the linear collision operator's structure and establishing estimates for the nonlinear terms under unequal mass conditions. Consequently, these results may help advance spectral analysis for soft potentials as well as $L^2,L^{\infty}$ frameworks in future studies of multi-component Boltzmann equations.