Decently regular steady solutions to the compressible NSAC system

Abstract

We aim at proving existence of weak solutions to the stationary compressible Navier-Stokes system coupled with the Allen-Cahn equation - NSAC system. 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 $\gamma$ \hbox{$ (p \sim \varrho^\gamma$)}. As a corollary we obtain weak solutions for a less restrictive case losing pointwise boundedness of the density.