The Boltzmann Equation for Uniform Shear Flow

The uniform shear flow for 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 strays 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. This 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.

Automated summary

The study examines the uniform shear flow of rarefied gas and identifies the reason for temperature increase over time, as well as the characteristics of self-similar profiles. The non-negativity of these profiles is justified through large time asymptotic stability.