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Analysis of an Incompressible Navier–Stokes–Maxwell–Stefan System

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Abstract

The Maxwell–Stefan equations for the molar fluxes, supplemented by the incompressible Navier–Stokes equations governing the fluid velocity dynamics, 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 different molar masses of the fluid components. 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.

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Authors and Affiliations

  1. School of Sciences, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Xiuqing Chen

  2. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040, Vienna, Austria

    Ansgar Jüngel

Authors
  1. Xiuqing Chen
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  2. Ansgar Jüngel
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Correspondence to Ansgar Jüngel.

Additional information

Communicated by L. Caffarelli

The first author acknowledges support from the National Natural Science Foundation of China, Grants 11101049 and 11471050. The second author was partially supported by the Austrian Science Fund (FWF), Grants P22108, P24304, P27352, I395, and W1245, and the Austrian–French Project of the Austrian Exchange Service (ÖAD).

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Chen, X., Jüngel, A. Analysis of an Incompressible Navier–Stokes–Maxwell–Stefan System. Commun. Math. Phys. 340, 471–497 (2015). https://doi.org/10.1007/s00220-015-2472-z

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  • DOI: https://doi.org/10.1007/s00220-015-2472-z

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