# Two-component viscous ﬂows with density-dependent transition layer

### Gabriele Witterstein

Technische Universität München, München Garching, Germany

## Abstract

We present a model describing ﬂows which undergo a phase change caused by chemical reactions in the material ﬂow. In mathematical terms, we consider a free boundary problem for the onedimensional, compressible Navier–Stokes system coupled with an Allen–Cahn equation. One free boundary arises at the boundary of the ﬂow to a vacuum state with a jump in the density. Another approximate free boundary arises due to the Allen–Cahn equation modeling the phase transition of the ﬂow. We consider a density-dependent viscosity $µ=ρ_{θ}$ and a density-dependent transition layer of thickness $δ=δ~ρ_{−1}$. We establish the existence of a unique weak global solution, provided that $θ<1/3$.

## Cite this article

Gabriele Witterstein, Two-component viscous ﬂows with density-dependent transition layer. Interfaces Free Bound. 11 (2009), no. 4, pp. 537–568

DOI 10.4171/IFB/222