### Decently regular steady solutions to the compressible NSAC system

DOI: http://dx.doi.org/10.12775/TMNA.2016.042

#### 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.

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.

#### Keywords

Navier--Stokes--Allen--Cahn equations; steady compressible flow; existence and regularity of the solutions

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