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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

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.

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. BoudinB. 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. BoudinB. 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. BoudinB. 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. DausA. 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. DesvillettesTh. 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. HerbergM. MeyriesJ. 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. ShigesadaK. 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. ThirietD. DouguetJ.-C. BonnetC. 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. BoudinB. 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. BoudinB. 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. BoudinB. 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. DausA. 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. DesvillettesTh. 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. HerbergM. MeyriesJ. 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. ShigesadaK. 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. ThirietD. DouguetJ.-C. BonnetC. 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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