Convergence to Equilibrium for the Linearized Cometary Flow Equation

We study convergence to equilibrium for certain spatially inhomogenous kinetic equations, such as discrete velocity models or a linearization of a kinetic model for cometary flow. For such equations, the convergence to a unique equilibrium state is the result of, firstly, the dissipative effects of the collision operator, which morphs the solution towards an entropy minimizing local equilibrium state, and secondly, the transport operator as well as the imposed periodic boundary conditions, which repulse the solution from the set of local equilibria as long as the approached local equilibrium is not the global one. This behaviour is quantified in a system of differential inequalities of relative entropies with respect to different (sub)classes of local equilibria, respectively, the global equilibrium. We introduce projection operators leading to a convenient notation.