Asymptotic behaviour of solutions to the Navier-Stokes equations of a two-dimensional compressible flow

In this paper, we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p(ϱ) = αϱlog ^d (ϱ) for large ϱ. Here d > 2, a > 0. We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is also briefly discussed.