Inelastic Boltzmann equation under shear heating

TL;DR

This paper analyzes the inelastic Boltzmann equation with shear heating, establishing existence of self-similar profiles and describing the long-term temperature behavior depending on the balance of heating and cooling effects.

Contribution

It introduces the existence of non-Maxwellian self-similar profiles under small deformation and nearly elastic conditions, and characterizes the asymptotic temperature behavior.

Findings

Temperature diverges when shear heating dominates.

Temperature vanishes when inelastic cooling dominates.

Temperature stabilizes to a positive constant when effects are balanced.

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.