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Non-homogeneous Navier-Stokes systems with order-parameter dependent stresses
Helmut Abels and Yutaka Terasawa
We consider the Navier-Stokes system with variable density and variable viscosity coupled to a transport equation for an order parameter $c$. Moreover, an extra stress depending on $c$ and $\nabla c$, which describes surface tension like effects, is included in the Navier-Stokes system. Such a system arises e.g. for certain models for granular flows and as a diffuse interface model for a two-phase flow of viscous incompressible fluids. The so-called density-dependent Navier-Stokes system is also a special case of our system. We prove short-time existence of strong solution in $L^q$-Sobolev spaces with $q>d$. We consider the case of a bounded domain and an asymptotically flat layer with combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system.