Decently regular steady solutions to the compressible NSAC system. (English) Zbl 1365.76274

Axmann, Šimon; Mucha, Piotr Bogusław

Topol. Methods Nonlinear Anal. 48, No. 1, 131-157 (2016).

Summary: 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 \(\gamma\,(p\sim\varrho^\gamma)\). As a corollary we obtain weak solutions for a less restrictive case losing pointwise boundedness of the density.

MSC:

76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30	Navier-Stokes equations

Keywords:

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