Global–in–time existence for liquid mixtures subject to a generalised incompressibility constraint

• Published in: Journal of mathematical analysis and applications Vol. 499; no. 2; p. 125059
• Main Author: Druet, Pierre-Etienne
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.07.2021
• Subjects: Complex fluid; Defect measure; Fluid mixture; Global weak solutions; Hard-sphere pressure law; Incompressible fluid; Low Mach-number; Mathematical analysis; Multicomponent flow; Multicomponent flow; Hard-sphere pressure law; Complex fluid; Mathematical analysis; Global weak solutions; Defect measure; Fluid mixture; Incompressible fluid; Low Mach-number
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2021.125059

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.