Moment estimates and well-posedness of the binary-ternary Boltzmann equation

Abstract

In this paper, we show global well-posedness as well as the generation and propagation of polynomial and exponential moments for the binary-ternary Boltzmann equation with integrable angular kernel. We show that the homogeneous binary-ternary equation behaves like the better of the two - the purely binary or purely ternary equation - in the sense that it exhibits the generation of moments corresponding to the part of the kernel with the highest potential rate. An important consequence of this is establishing, for the first time, generation of moments of solutions even if one of the potentials corresponds to Maxwell molecules (something that is not known for purely binary or purely ternary equation with Maxwell type interactions). To address these questions, we develop compact manifold angular averaging estimates for the ternary collision operator. This is the first paper which discusses this type of questions for the binary-ternary Boltzmann equation and opens the door for studying moments properties of gases with higher collisional density.