2019, Druet, Pierre-Étienne

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.

2020, Bothe, Dieter, Druet, Pierre-Étienne

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.