Inelastic Boltzmann equation under shear heating
José A. Carrillo, Kam Fai Chan, Renjun Duan, Zongguang Li
TL;DR
This work analyzes the spatially homogeneous inelastic Boltzmann equation under shear heating, balancing heating and cooling to understand long-time behavior. It develops a unified framework using both Radon-measure and Fourier-space analyses, employing a self-similar scaling to seek stationary profiles. The authors prove the existence of non-Maxwellian self-similar profiles in both Radon and Fourier settings under small deformation and near-elastic restitution and establish convergence of global solutions to these profiles with rates measured in Toscani-type metrics. They also characterize temperature dynamics under uniform shear flow, identifying regimes where heating dominates, cooling dominates, or the two effects balance, with the balanced case corresponding to the stationary self-similar state. The results advance rigorous understanding of shear-driven granular gas dynamics and provide tools for quantitative description of non-equilibrium self-similar states.
Abstract
In this paper, we study the spatially homogeneous inelastic Boltzmann equation for the angular cutoff pseudo-Maxwell molecules with an additional term of linear deformation. We establish the existence of non-Maxwellian self-similar profiles under the assumption of small deformation in the nearly elastic regime, and also obtain weak convergence to these self-similar profiles for global-in-time solutions with initial data that have finite mass and finite $p$-th order moment for any $2<p\leq 4$. Our results confirm the competition between shear heating and inelastic cooling that governs the large time behavior of temperature. Specifically, temperature increases to infinity if shear heating dominates, decreases to zero if inelastic cooling prevails, and converges to a positive constant if the two effects are balanced. In the balanced scenario, the corresponding self-similar profile aligns with the steady solution.