# American Institute of Mathematical Sciences

## A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion

 Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany

Received  March 2020 Revised  August 2020 Published  November 2020

Fund Project: The author was supported by the grant D1117/1-1 of the German Science Foundation (DFG)

After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic–hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids ex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

Citation: Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020458
##### References:
 [1] D. Bothe and P.-E. Druet, The free energy of incompressible liquid mixtures: some mathematical insights, In preparation. Google Scholar [2] D. Bothe and P.-E. Druet, Mass transport in multicomponent compressible fluids: local and global well-posedness in classes of strong solutions for general class-one models, 2019. Available at http://www.wias-berlin.de/preprint/2658/wias_preprints_2658.pdf, and at arXiv: 2001.08970 [math.AP]. Google Scholar [3] D. Bothe and P.-E. Druet, On the structure of continuum thermodynamical diffusion fluxes: a novel closure scheme and its relation to the Maxwell-Stefan and the Fick-Onsager approach, 2020. Available at http://www.wias-berlin.de/preprint/2749/wias_preprints_2749.pdf, and arXiv: 2008.05327 [math-ph]. Google Scholar [4] D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757-1805.  doi: 10.1007/s00707-014-1275-1.  Google Scholar [5] D. Bothe and J. Prüss, Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition – the isothermal incompressible case, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 673-696.  doi: 10.3934/dcdss.2017034.  Google Scholar [6] R. Brdicka, Grundlagen der physikalischen Chemie, Deutscher Verlag der Wissenschaften, Berlin, 1981. Google Scholar [7] X. Chen and A. Jüngel, Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system, Commun. Math. Phys., 340 (2015), 471-497.  doi: 10.1007/s00220-015-2472-z.  Google Scholar [8] W. Dreyer, P.-E. Druet, P. Gajewski and C. Guhlke, Existence of weak solutions for improved Nernst-Planck-Poisson models of compressible reacting electrolytes, Z. Angew. Math. Phys., 71 (2020), Paper No. 119, 68 pp. Open access: https://doi.org/10.1007/s00033-020-01341-5. doi: 10.1007/s00033-020-01341-5.  Google Scholar [9] P.-E. Druet, Analysis of improved Nernst-Planck-Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix, Preprint 2321 of the WIAS, 2016. Available at http://www.wias-berlin.de/preprint/2321/wias_preprints_2321.pdf. Google Scholar [10] P.-E. Druet, Global–in–time existence for liquid mixtures subject to a generalised incompressibility constraint, Preprint 2622 of the WIAS, 2019. Available at http://www.wias-berlin.de/preprint/2622/wias_preprints_2622.pdf. Google Scholar [11] E. Feireisl, A. Novotnỳ and H. Petzeltovà, On the existence of globally defined weak solutions to the Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar [12] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1580-6.  Google Scholar [13] M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass–action kinetics, Nonlinear Analysis: Theory, Methods & Applications, 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.  Google Scholar [14] P.-L. Lions, Mathematical Topics in Fluid Dynamics. Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998.  Google Scholar [15] Ladyzenskaja, Solonnikov and Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Volume 23 of Translations of mathematical monographs, AMS, 1968. Google Scholar [16] P.B. Mucha, M. Pokorny and E. Zatorska, Heat-conducting, compressible mixtures with multicomponent diffusion: Construction of a weak solution, SIAM J. Math. Anal., 47 (2015), 3747-3797.  doi: 10.1137/140957640.  Google Scholar [17] M. Marion and R. Temam, Global existence for fully nonlinear reaction-diffusion systems describing multicomponent reactive flows, J. Math. Pures Appl., 104 (2015), 102-138.  doi: 10.1016/j.matpur.2015.02.003.  Google Scholar [18] T. Piasecki, Y. Shibata and E. Zatorska, On strong dynamics of compressible two-component mixture flow, SIAM J. Math. Anal., 51 (2019), 2793-2849.  doi: 10.1137/17M1151134.  Google Scholar

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##### References:
 [1] D. Bothe and P.-E. Druet, The free energy of incompressible liquid mixtures: some mathematical insights, In preparation. Google Scholar [2] D. Bothe and P.-E. Druet, Mass transport in multicomponent compressible fluids: local and global well-posedness in classes of strong solutions for general class-one models, 2019. Available at http://www.wias-berlin.de/preprint/2658/wias_preprints_2658.pdf, and at arXiv: 2001.08970 [math.AP]. Google Scholar [3] D. Bothe and P.-E. Druet, On the structure of continuum thermodynamical diffusion fluxes: a novel closure scheme and its relation to the Maxwell-Stefan and the Fick-Onsager approach, 2020. Available at http://www.wias-berlin.de/preprint/2749/wias_preprints_2749.pdf, and arXiv: 2008.05327 [math-ph]. Google Scholar [4] D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757-1805.  doi: 10.1007/s00707-014-1275-1.  Google Scholar [5] D. Bothe and J. Prüss, Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition – the isothermal incompressible case, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 673-696.  doi: 10.3934/dcdss.2017034.  Google Scholar [6] R. Brdicka, Grundlagen der physikalischen Chemie, Deutscher Verlag der Wissenschaften, Berlin, 1981. Google Scholar [7] X. Chen and A. Jüngel, Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system, Commun. Math. Phys., 340 (2015), 471-497.  doi: 10.1007/s00220-015-2472-z.  Google Scholar [8] W. Dreyer, P.-E. Druet, P. Gajewski and C. Guhlke, Existence of weak solutions for improved Nernst-Planck-Poisson models of compressible reacting electrolytes, Z. Angew. Math. Phys., 71 (2020), Paper No. 119, 68 pp. Open access: https://doi.org/10.1007/s00033-020-01341-5. doi: 10.1007/s00033-020-01341-5.  Google Scholar [9] P.-E. Druet, Analysis of improved Nernst-Planck-Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix, Preprint 2321 of the WIAS, 2016. Available at http://www.wias-berlin.de/preprint/2321/wias_preprints_2321.pdf. Google Scholar [10] P.-E. Druet, Global–in–time existence for liquid mixtures subject to a generalised incompressibility constraint, Preprint 2622 of the WIAS, 2019. Available at http://www.wias-berlin.de/preprint/2622/wias_preprints_2622.pdf. Google Scholar [11] E. Feireisl, A. Novotnỳ and H. Petzeltovà, On the existence of globally defined weak solutions to the Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar [12] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1580-6.  Google Scholar [13] M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass–action kinetics, Nonlinear Analysis: Theory, Methods & Applications, 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.  Google Scholar [14] P.-L. Lions, Mathematical Topics in Fluid Dynamics. Vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998.  Google Scholar [15] Ladyzenskaja, Solonnikov and Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Volume 23 of Translations of mathematical monographs, AMS, 1968. Google Scholar [16] P.B. Mucha, M. Pokorny and E. Zatorska, Heat-conducting, compressible mixtures with multicomponent diffusion: Construction of a weak solution, SIAM J. Math. Anal., 47 (2015), 3747-3797.  doi: 10.1137/140957640.  Google Scholar [17] M. Marion and R. Temam, Global existence for fully nonlinear reaction-diffusion systems describing multicomponent reactive flows, J. Math. Pures Appl., 104 (2015), 102-138.  doi: 10.1016/j.matpur.2015.02.003.  Google Scholar [18] T. Piasecki, Y. Shibata and E. Zatorska, On strong dynamics of compressible two-component mixture flow, SIAM J. Math. Anal., 51 (2019), 2793-2849.  doi: 10.1137/17M1151134.  Google Scholar
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