Two-component viscous flows with density-dependent transition layer

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 $\mu ={\rho }^{\theta }$ and a density-dependent transition layer of thickness $\delta =\tilde{\delta }{\rho }^{-1}$. We establish the existence of a unique weak global solution, provided that $\theta <1/3$.