Decently regular steady solutions to the compressible NSAC system

TL;DR

This paper proves the existence of weak solutions with bounded densities for a coupled compressible Navier-Stokes and Allen-Cahn system in a bounded domain, under certain conditions on the heat capacity ratio.

Contribution

It establishes the existence of weak solutions to a coupled PDE system with bounded densities, extending previous results to more general conditions.

Findings

Existence of weak solutions with bounded densities.

Weak solutions exist for large heat capacity ratio γ.

Weak solutions also exist without point-wise boundedness of density.

Abstract

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 $\gamma$ ($p \sim \rho^\gamma$). As a corollary we obtain weak solutions for a less restrictive case losing point-wise boundedness of the density.