Non-homogeneous Navier-Stokes systems with order-parameter-dependent stresses. (English) Zbl 1303.76008
Summary: 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 of 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 a 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.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35Q30 | Navier-Stokes equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76T25 | Granular flows |
Keywords:
Navier-Stokes equations; free boundary value problems; maximal regularity; diffuse interface models; granular flows; non-stationary Stokes systemReferences:
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