Non-homogeneous Navier-Stokes systems with order-parameter dependent stresses
Helmut Abels and Yutaka Terasawa
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Submission date: 30. Mar. 2009
published in: Mathematical methods in the applied sciences, 33 (2010) 13, p. 1532-1544
DOI number (of the published article): 10.1002/mma.1264
MSC-Numbers: 76D05, 35Q30, 35R35, 76T99, 76D27, 76D45
Keywords and phrases: Navier Stokes equations, free boundary value problems, maximal regularity, diffuse interface models, granular flows, non-stationary Stokes system
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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 , 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 -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.