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A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior
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 |
Recently, the authors proved [
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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