Well-posedness analysis of multicomponent incompressible flow models
- Bothe, Dieter
- Druet, Pierre-Étienne
2010 Mathematics Subject Classification
- 35M33 35Q30 76N10 35D35 35B65 35B35 35K57 35Q35 35Q79 76R50 80A17 80A32 92E20
- Multicomponent flow, complex fluid, fluid mixture, incompressible fluid, low Mach-number, strong solutions
In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the Navier--Stokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.