Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint
- Druet, Pierre-Étienne
2010 Mathematics Subject Classification
- 35M33 35Q30 76N10 76D05 35D30 35B65
- Multicomponent flow, complex fluid, fluid mixture, incompressible fluid, low Mach-number, mathematical analysis, global weak solutions, hard-sphere pressure law, defect measure
We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L1. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.