The Boltzmann equation for uniform shear flow

TL;DR

This paper analyzes the uniform shear flow in a rarefied gas using the Boltzmann equation, establishing existence, regularity, and stability of self-similar profiles, especially for Maxwell molecules at small shear rates.

Contribution

It provides the first rigorous proof of existence, regularity, and stability of self-similar solutions for the Boltzmann equation under shear flow, including non-negativity and tail behavior.

Findings

Existence and uniqueness of self-similar profiles for Maxwell molecules.

Profiles exhibit algebraic high-velocity tails.

Exponential convergence rates to self-similar profiles.

Abstract

The uniform shear flow for the rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat and the system becomes far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous perturbation framework, and the exponential rates of convergence are also obtained with the size proportional to the second order shear rate. The analysis supports the numerical result that the self-similar profile admits an algebraic high-velocity tail that is the key difficulty to overcome in the proof.