The Boltzmann equation for uniform shear flow

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. Contents 1. Intoduction 1 2. Basic estimates 10 3. Large velocity decay of K 13 4. Steady problem 17 5. Local Existence 30 6. Convergence to the steady state 34 References 50

The Boltzmann equation for plane Couette flow

“…as shown in the proof. The result is consistent with the one in [14] for uniform shear flow without boundaries in the spatially homogeneous setting.…”

“…The current study of the plane Couette flow with boundaries is motivated by the previous work [14] by the first two authors for uniform shear flow via the Boltzmann equation without boundaries. We refer readers to [9,12,21,27,36,37] and references therein for more details of the topic on uniform shear flow.…”

“…But it remains unsolved how to justify the large time asymptotics toward the stationary solution for the time-dependent problem in the same setting of the fluid limit. In this paper, motivated by [14], instead of constructing the fluid dynamic approximation solutions, we focus on the existence and dynamical stability of the plane Couette flow with the finite Knudsen number for both the steady and unsteady problems.…”

“…This means that the solution to the Couette flow problem (1.1), (1.2) and (1.3) is established around the local Maxwellian µ(v x − αy, v y , v z ) instead of the global Maxwellian µ such that the kinetic diffusive reflection boundary condition (1.2) is satisfied for the background solution µ(v x − αy, v y , v z ). In addition, as mentioned before, it seems more convenient to use the formulation with shear forces than the original one driven by the relative motion of boundaries in order to understand the asymptotic behavior of solutions in the limit L → ∞, that is, how the Couette flow with boundaries converges to a shear flow without boundary that is closely related to what has been studied in the previous works [14] for uniform shear flow in the spatially homogeneous setting.…”

“…The form (1.24) is then consistent with the uniform shear flow in [14]. To solve the boundary value problem (1.11) and (1.12), the same approach as for treating the remainder G R can be applied.…”