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Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses
Author(s) -
Abels Helmut,
Terasawa Yutaka
Publication year - 2010
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1264
Subject(s) - mathematics , mathematical analysis , bounded function , dirichlet boundary condition , sobolev space , domain (mathematical analysis) , surface tension , variable (mathematics) , navier–stokes equations , weak solution , boundary value problem , compressibility , boundary (topology) , stokes flow , viscosity , flow (mathematics) , geometry , mechanics , physics , thermodynamics
Abstract 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 ∇ 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. Copyright © 2010 John Wiley & Sons, Ltd.