Cross-Diffusion Systems with Entropy Structure

Jüngel, Ansgar

  • Proceedings of Equadiff 14, Publisher: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing(Bratislava), page 181-190

Abstract

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Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak solutions is recalled, and some open problems are presented.

How to cite

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Jüngel, Ansgar. "Cross-Diffusion Systems with Entropy Structure." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 181-190. <http://eudml.org/doc/294882>.

@inProceedings{Jüngel2017,
abstract = {Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak solutions is recalled, and some open problems are presented.},
author = {Jüngel, Ansgar},
booktitle = {Proceedings of Equadiff 14},
keywords = {Strongly coupled parabolic systems, local existence of solutions, global existence of solutions, gradient flow, duality method, boundedness-by-entropy method, nonlinear Aubin-Lions lemma, Kullback-Leibler entropy},
location = {Bratislava},
pages = {181-190},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {Cross-Diffusion Systems with Entropy Structure},
url = {http://eudml.org/doc/294882},
year = {2017},
}

TY - CLSWK
AU - Jüngel, Ansgar
TI - Cross-Diffusion Systems with Entropy Structure
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 181
EP - 190
AB - Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak solutions is recalled, and some open problems are presented.
KW - Strongly coupled parabolic systems, local existence of solutions, global existence of solutions, gradient flow, duality method, boundedness-by-entropy method, nonlinear Aubin-Lions lemma, Kullback-Leibler entropy
UR - http://eudml.org/doc/294882
ER -

References

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  1. Alt, H.-W., Luckhaus., S., Quasilinear elliptic-parabolic differential equations, . Math. Z. 183(1983), 311-341. MR0706391
  2. Amann., H., Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, . Diff. Int. Eqs. 3 (1990), 13-75. MR1014726
  3. Amann., H., Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, . In: H.J. Schmeisser and H. Triebel (editors), Function Spaces, Differential Operators and Nonlinear Analysis, pp. 9-126. Teubner, Stuttgart, 1993. MR1242579
  4. Bothe., D., On the Maxwell-Stefan equations to multicomponent diffusion, . In: Progress in Nonlinear Differential Equations and their Applications, pp. 81-93. Springer, Basel, 2011. MR3052573
  5. Burger, M., Francesco, M. Di, Pietschmann, J.-F., HASH(0x1502c88), Schlake., B., Nonlinear cross-diffusion with size exclusion, . SIAM J. Math. Anal. 42 (2010), 2842-2871. MR2745794
  6. Carrillo, J. A., Jüngel, A., Markowich, P., Toscani, G., HASH(0x15056a0), Unterreiter., A., Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, . Monatsh. Math. 133 (2001), 1-82. MR1853037
  7. Chen, L., Jüngel., A., Analysis of a parabolic cross-diffusion population model without selfdiffusion, . J. Diff. Eqs. 224 (2006), 39-59. MR2220063
  8. Chen, X., Daus, E., HASH(0x1507b50), Jüngel., A., Global existence analysis of cross-diffusion population systems for multiple species, . To appear in Arch. Ration. Mech. Anal., 2017. arXiv:1608.03696. MR3740386
  9. Chen, X., Jüngel., A., A note on the uniqueness of weak solutions to a class of cross-diffusion systems, . Submitted for publication, 2017. arXiv:1706.08812. MR3820423
  10. Chen, X., Jüngel., A., Global renormalized solutions to reaction-cross-diffusion systems, . Work in progress, 2017. MR3996789
  11. Chen, X., Jüngel, A., HASH(0x150e390), Liu., J.-G., A note on Aubin-Lions-Dubinskiı̆ lemmas, . Acta Appl. Math. 133 (2014), 33-43. MR3255076
  12. Desvillettes, L., Lepoutre, T., HASH(0x150edb0), Moussa., A., Entropy, duality, and cross diffusion, . SIAM J. Math. Anal. 46 (2014), 820-853. MR3165911
  13. Desvillettes, L., Lepoutre, T., Moussa, A., HASH(0x1511518), Trescases., A., On the entropic structure of reactioncross diffusion systems, . Commun. Partial Diff. Eqs. 40 (2015), 1705-1747. MR3359162
  14. Dı́az, J. I., Galiano, G., HASH(0x1512ff0), Jüngel., A., On a quasilinear degenerate system arising in semiconductor theory. Part I: existence and uniqueness of solutions, . Nonlin. Anal. RWA 2 (2001), 305-336. MR1835610
  15. Dreher, M., Jüngel., A., Compact families of piecewise constant functions in L p ( 0 , T ; B ) , . Nonlin. Anal. 75 (2012), 3072-3077. MR2890969
  16. Fellner, K., Prager, W., Tang., B. Q., The entropy method for reaction-diffusion systems without detailed balance: first order chemical reaction networks, . Kinetic Related Models 10 (2017), 1055-1087. MR3622100
  17. Fischer., J., Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, . Arch. Ration. Mech. Anal. 218 (2015), 553-587. MR3360745
  18. Fischer., J., Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, . Submitted for publication, 2017. arXiv:1703.00730. MR3659829
  19. Gajewski., H., On a variant of monotonicity and its application to differential equations, . Nonlin. Anal. TMA 22 (1994), 73-80. MR1256171
  20. Gerstenmayer, A., Jüngel., A., Analysis of a degenerate parabolic cross-diffusion system for ion transport, . Submitted for publication, 2017. arXiv:1706.07261. MR3759555
  21. Herberg, M., Meyries, M., Prüss, J., HASH(0x151c4d8), Wilke., M., Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, . Nonlin. Anal. 159 (2017), 264-284. MR3659831
  22. Hittmeir, S., Jüngel., A., Cross diffusion preventing blow up in the two-dimensional Keller–Segel model, . SIAM J. Math. Anal. 43 (2011), 997-1022. MR2801186
  23. Hoang, L., Nguyen, T., Phan., T. V., Gradient estimates and global existence of smooth solutions to a cross-diffusion system, . SIAM J. Math. Anal. 47 (2015), 2122-2177. MR3352604
  24. Jüngel., A., The boundedness-by-entropy method for cross-diffusion systems, . Nonlinearity 28(2015), 1963-2001. MR3350617
  25. Jüngel., A., Entropy Methods for Diffusive Partial Differential Equations, . BCAM Springer Briefs,2016. MR3497125
  26. Jüngel, A., Stelzer., I., Entropy structure of a cross-diffusion tumor-growth model, . Math. Models Meth. Appl. Sci. 22 (2012), 1250009, 26 pages. MR2924784
  27. Jüngel, A., Zamponi., N., Qualitative behavior of solutions to cross-diffusion systems from population dynamics, . J. Math. Anal. Appl. 440 (2016), 794-809. MR3484995
  28. Kullback, S., Leibler., R., On information and sufficiency, . Ann. Math. Statist. 22 (1951), 79-86. MR0039968
  29. Le., D., Global existence for a class of strongly coupled parabolic systems, . Ann. Matem. 185 (2006), 133-154. MR2179585
  30. Lepoutre, T., Moussa., A., Entropic structure and duality for multiple species cross-diffusion systems, . Nonlin. Anal. 159 (2017), 298-315. MR3659833
  31. Moussa., A., Some variants of the classical Aubin-Lions lemma, . J. Evol. Eqs. 16 (2016), 65-93. MR3466213
  32. Otto., F., L 1 -contraction and uniqueness for quasilinear elliptic-parabolic equations, . J. Diff. Eqs. 131 (1996), 20-38. MR1415045
  33. Pham, D., Temam., R., A result of uniqueness of solutions of the Shigesada-Kawasaki-Teramoto equations, . Adv. Nonlin. Anal., online first, 2017. arXiv:1703.10544. MR3918387
  34. Pierre., M., Global existence in reaction-diffusion systems with control of mass: a survey, . Milan J. Math. 78 (2010), 417-455. MR2781847
  35. Pierre, M., Schmitt., D., Blow up in reaction-diffusion systems with dissipation of mass, . SIAM J. Math. Anal. 28 (1997), 259-269. MR1434034
  36. Shigesada, N., Kawasaki, K., HASH(0x1532dc8), Teramoto., E., Spatial segregation of interacting species, . J. Theor. Biol. 79 (1979), 83-99. MR0540951
  37. Stará, J., John., O., Some (new) counterexamples of parabolic systems, . Comment. Math. Univ. Carolin. 36 (1995), 503-510. MR1364491
  38. Wesselingh, J., Krishna., R., Mass Transfer in Multicomponent Mixtures, . Delft University Press, Delft, 2000. 
  39. Zamponi, N., Jüngel., A., Analysis of degenerate cross-diffusion population models with volume filling, . Ann. Inst. H. Poincaré Anal. Non Lin. 34 (2017), 1-29. MR3592676

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