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Decently regular steady solutions to the compressible NSAC system

We aim at proving existence of weak solutions to the stationary compressible Navier-Stokes system coupled with the Allen-Cahn equation. The model is studied in a bounded three dimensional domain with slip boundary conditions for the momentum equations and the Neumann condition for the Allen-Cahn model. The main result establishes existence of weak solutions with bounded densities. The construction is possible assuming sufficiently large value of the heat capacity ratio $γ$ ($p \sim ρ^γ$). As a corollary we obtain weak solutions for a less restrictive case losing point-wise boundedness of the density.