Stability analysis of the plane Couette flow for a model kinetic equation

The stability of the plane Couette flow is studied using the simplified Boltzmann equation (the BGK equation) in which the high modes in the space of velocities and coordinates are truncated. The solution to the Navier-Stokes equation with small additional terms depending on the Knudsen number is used as the stationary solution. We assume that the perturbations depend only on the coordinate that is orthogonal to the flow. The density perturbations are assumed to be nonzero. In this approximation, the problem is found to be unstable in the case of small Knudsen numbers.