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June  2022, 42(6): 2747-2773. doi: 10.3934/dcds.2021210

Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

Andrea Bondesan 1, and  Marc Briant 2,,

1. 

University of Graz, Institute of Mathematics and Scientific Computing, 8010 Graz, Austria
2. 

Université de Paris, Laboratoire MAP5, UMR CNRS 8145, F-75006 Paris, France

*Corresponding author

Received  January 2021 Revised  October 2021 Published  June 2022 Early access  January 2022

Fund Project: The authors would like to thank Laurent Boudin and Bérénice Grec for fruitful discussions on the theory of gaseous and fluid mixtures

Recently, the authors proved [2] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [2]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.

Keywords: Fluid mixtures, incompressible Maxwell-Stefan, perturbative Cauchy theory.
Mathematics Subject Classification: Primary: 35A01, 76B03, 35Q35; Secondary: 35K51, 35K55.
Citation: Andrea Bondesan, Marc Briant. Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2747-2773. doi: 10.3934/dcds.2021210
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, H.J. Schmeisser, H. Triebel (eds), Teubner, Stuttgart, Leipzig, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.

[2]

A. Bondesan and M. Briant, Stability of the Maxwell-Stefan system in the diffusion asymptotics of the Boltzmann multi-species equation, Comm. Math. Phys., 382 (2021), 381-440.  doi: 10.1007/s00220-021-03976-5.

[3]

D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer Basel AG, Basel, 80 (2011), 81-93.  doi: 10.1007/978-3-0348-0075-4_5.

[4]

L. Boudin, D. Götz and B. Grec, Diffusion models of multicomponent mixtures in the lung, CEMRACS 2009: Mathematical Modelling in Medicine, ESAIM Proc., EDP Sci., Les Ulis, 30 (2010), 90–103. doi: 10.1051/proc/2010008.

[5]

L. Boudin, B. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Anal., 159 (2017), 40-61.  doi: 10.1016/j.na.2017.01.010.

[6]

L. Boudin, B. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440.  doi: 10.3934/dcdsb.2012.17.1427.

[7]

L. Boudin, B. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Acta Applicandae Mathematicae, 136 (2015), 79-90.  doi: 10.1007/s10440-014-9886-z.

[8]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[9]

H. Chang, Multicomponent diffusion in the lung, Fed. Proc., 39 (1980), 2759-2764. 

[10] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, Cambridge, 1990. 
[11]

X. Chen and A. Jüngel, Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system, Comm. Math. Phys., 340 (2015), 471-497.  doi: 10.1007/s00220-015-2472-z.

[12]

E. S. Daus, A. Jüngel and B. Q. Tang, Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type, Archive Rat. Mech. Anal., 235 (2020), 1059-1104.  doi: 10.1007/s00205-019-01439-9.

[13]

L. Desvillettes, Th. Lepoutre and A. Moussa, Entropy, duality, and cross diffusion, SIAM J. Math. Anal., 46 (2014), 820-853.  doi: 10.1137/130908701.

[14]

A. Fick, Ueber diffusion, Ann. Der Physik, 170 (1855), 59-86.  doi: 10.1002/andp.18551700105.

[15]

V. Giovangigli, Mass conservation and singular multicomponent diffusion algorithms, Numerical Combustion, 351 (2005), 310-322.  doi: 10.1007/3-540-51968-8_94.

[16]

V. Giovangigli, Multicomponent Flow Modeling, Modeling and Simulation in Science, Engineering and Technology, Birkhüser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.

[17]

V. Giovangigli and M. Massot, Les mélanges gazeux réactifs. I. Symétrisation et existence locale, C. R. Acad. Sci. Paris, 323 (1996), 1153-1158. 

[18]

M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Anal., 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.

[19]

H. Hutridurga and F. Salvarani, On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases, Math. Meth. in Appl. Sci., 40 (2017), 803-813.  doi: 10.1002/mma.4013.

[20]

H. Hutridurga and F. Salvarani, Existence and uniqueness analysis of a non-isothermal cross-diffusion system of Maxwell-Stefan type, Appl. Math. Lett., 75 (2018), 108-113.  doi: 10.1016/j.aml.2017.06.007.

[21]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.

[22]

A. Jüngel and I. V. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, SIAM J. Math. Anal., 45 (2013), 2421-2440.  doi: 10.1137/120898164.

[23]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto Univ., https://ci.nii.ac.jp/naid/10026415245/en/.

[24]

R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci., 52 (1997), 861-911.  doi: 10.1016/S0009-2509(96)00458-7.

[25]

Y. Lou and S. Martínez, Evolution of cross-diffusion and self-diffusion, J. Biol. Dyn., 3 (2009), 410-429.  doi: 10.1080/17513750802491849.

[26]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.

[27]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.

[28]

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.

[29]

J. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. Lond., 157 (1867), 49–88, http://dx.doi.org/10.1098/rstl.1867.0004.

[30]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[31]

J. Stefan, Über das gleichgewicht und die bewegung, insbesondere die diffusion von gasgemengen, Akad. Wiss. Wien, 63 (1871), 63-124. 

[32]

M. Thiriet, D. Douguet, J.-C. Bonnet, C. Canonne and C. Hatzfeld, The effect on gas mixing of a He-O2 mixture in chronic obstructive lung diseases, Bull. Eur. Physiopathol. Respir., 15 (1979), 1053-1068. 

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, H.J. Schmeisser, H. Triebel (eds), Teubner, Stuttgart, Leipzig, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.

[2]

A. Bondesan and M. Briant, Stability of the Maxwell-Stefan system in the diffusion asymptotics of the Boltzmann multi-species equation, Comm. Math. Phys., 382 (2021), 381-440.  doi: 10.1007/s00220-021-03976-5.

[3]

D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer Basel AG, Basel, 80 (2011), 81-93.  doi: 10.1007/978-3-0348-0075-4_5.

[4]

L. Boudin, D. Götz and B. Grec, Diffusion models of multicomponent mixtures in the lung, CEMRACS 2009: Mathematical Modelling in Medicine, ESAIM Proc., EDP Sci., Les Ulis, 30 (2010), 90–103. doi: 10.1051/proc/2010008.

[5]

L. Boudin, B. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Anal., 159 (2017), 40-61.  doi: 10.1016/j.na.2017.01.010.

[6]

L. Boudin, B. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440.  doi: 10.3934/dcdsb.2012.17.1427.

[7]

L. Boudin, B. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Acta Applicandae Mathematicae, 136 (2015), 79-90.  doi: 10.1007/s10440-014-9886-z.

[8]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[9]

H. Chang, Multicomponent diffusion in the lung, Fed. Proc., 39 (1980), 2759-2764. 

[10] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, Cambridge, 1990. 
[11]

X. Chen and A. Jüngel, Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system, Comm. Math. Phys., 340 (2015), 471-497.  doi: 10.1007/s00220-015-2472-z.

[12]

E. S. Daus, A. Jüngel and B. Q. Tang, Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type, Archive Rat. Mech. Anal., 235 (2020), 1059-1104.  doi: 10.1007/s00205-019-01439-9.

[13]

L. Desvillettes, Th. Lepoutre and A. Moussa, Entropy, duality, and cross diffusion, SIAM J. Math. Anal., 46 (2014), 820-853.  doi: 10.1137/130908701.

[14]

A. Fick, Ueber diffusion, Ann. Der Physik, 170 (1855), 59-86.  doi: 10.1002/andp.18551700105.

[15]

V. Giovangigli, Mass conservation and singular multicomponent diffusion algorithms, Numerical Combustion, 351 (2005), 310-322.  doi: 10.1007/3-540-51968-8_94.

[16]

V. Giovangigli, Multicomponent Flow Modeling, Modeling and Simulation in Science, Engineering and Technology, Birkhüser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.

[17]

V. Giovangigli and M. Massot, Les mélanges gazeux réactifs. I. Symétrisation et existence locale, C. R. Acad. Sci. Paris, 323 (1996), 1153-1158. 

[18]

M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Anal., 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.

[19]

H. Hutridurga and F. Salvarani, On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases, Math. Meth. in Appl. Sci., 40 (2017), 803-813.  doi: 10.1002/mma.4013.

[20]

H. Hutridurga and F. Salvarani, Existence and uniqueness analysis of a non-isothermal cross-diffusion system of Maxwell-Stefan type, Appl. Math. Lett., 75 (2018), 108-113.  doi: 10.1016/j.aml.2017.06.007.

[21]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.

[22]

A. Jüngel and I. V. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, SIAM J. Math. Anal., 45 (2013), 2421-2440.  doi: 10.1137/120898164.

[23]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto Univ., https://ci.nii.ac.jp/naid/10026415245/en/.

[24]

R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci., 52 (1997), 861-911.  doi: 10.1016/S0009-2509(96)00458-7.

[25]

Y. Lou and S. Martínez, Evolution of cross-diffusion and self-diffusion, J. Biol. Dyn., 3 (2009), 410-429.  doi: 10.1080/17513750802491849.

[26]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.

[27]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.

[28]

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.

[29]

J. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. Lond., 157 (1867), 49–88, http://dx.doi.org/10.1098/rstl.1867.0004.

[30]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[31]

J. Stefan, Über das gleichgewicht und die bewegung, insbesondere die diffusion von gasgemengen, Akad. Wiss. Wien, 63 (1871), 63-124. 

[32]

M. Thiriet, D. Douguet, J.-C. Bonnet, C. Canonne and C. Hatzfeld, The effect on gas mixing of a He-O2 mixture in chronic obstructive lung diseases, Bull. Eur. Physiopathol. Respir., 15 (1979), 1053-1068. 

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