Well-posedness analysis of multicomponent incompressible flow models

TL;DR

This paper analyzes the mathematical well-posedness of multicomponent incompressible fluid models, establishing local and global existence results for solutions under certain conditions, thus advancing the theoretical understanding of such complex fluid systems.

Contribution

It extends previous work to incompressible multicomponent flows, reformulates the PDE system to handle constraints, and proves well-posedness and existence results for these models.

Findings

Local-in-time well-posedness of strong solutions

Global existence near equilibrium solutions

Reformulation eliminates positivity and incompressibility constraints

Abstract

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 stays constant. In this type of models, the velocity field in the Navier-Stokes equations is not solenoidal 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.