Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system

TL;DR

This paper proves the global existence and exponential decay of solutions for a complex coupled Navier-Stokes and Maxwell-Stefan system modeling multicomponent gaseous mixtures, using novel entropy methods.

Contribution

It introduces a new entropy functional approach to handle the singular diffusion matrix and cross-diffusion terms in the coupled system.

Findings

Proved global-in-time existence of bounded weak solutions.

Established exponential decay to steady state.

Developed entropy-based techniques for complex coupled PDEs.

Abstract

The incompressible Navier-Stokes equations coupled to the Maxwell-Stefan relations for the molar fluxes are analyzed in bounded domains with no-flux boundary conditions. The system models the dynamics of a multicomponent gaseous mixture under isothermal conditions. The global-in-time existence of bounded weak solutions to the strongly coupled model and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to the singular Maxwell-Stefan diffusion matrix, the cross-diffusion terms, and the Navier-Stokes coupling. The key idea of the proof is the use of a new entropy functional and entropy variables, which allows for a proof of positive lower and upper bounds of the mass densities without the use of a maximum principle.