Stationary solutions of the Navier-Stokes-Fourier system in planar domains with impermeable boundary

The existence of weak solutions to the Navier-Stokes-Fourier system describing the stationary states of a compressible, viscous, and heat conducting fluid in bounded 2D-domains is shown under fairly general and physically relevant constitutive relations. The equation of state of a real fluid is considered, where the admissible range of density is confined to a bounded interval (hard sphere model). The transport coefficients depend on the temperature in a general way including both gases and liquids behavior. The heart of the paper are new a priori bounds resulting from Trudinger-Moser inequality.