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Uniqueness of strong solutions and weak–strong stability in a system of cross-diffusion equations

  • Judith Berendsen
  • Martin Burger
  • Virginie Ehrlacher
  • Jan-Frederik PietschmannEmail author
Article
  • 24 Downloads

Abstract

Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in Ehrlacher and Bakhta (ESAIM Math Model Numer Anal, 2017). Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of non-negative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak–strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions

Notes

Acknowledgements

The work of MB has been supported by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse. MB and JFP acknowledge support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster. VE acknowledges support by the ANR via the ANR JCJC COMODO project. VE and JFP are grateful to the DAAD/ANR for their support via the project 57447206. Furthermore, the authors would like to thank Robert Haller-Dintelmann (TU Darmstadt) for useful discussions. We would also like to thank the anonymous referee for his very useful comments and suggestions.

References

  1. 1.
    L. Alasio et al. Trend to Equilibrium for Systems with Small Cross-Diffusion, preprint arXiv:1906.08060, 2019.
  2. 2.
    H. Amann et al. Dynamic theory of quasilinear parabolic equations: Reaction-diffusion systems, Differential and Integral Equations, 3(1):13–75, 1990.MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. Breden, L. Desvillettes, K. Fellner Smoothness of moments of the solutions of discrete coagulation equations with diffusion, Monatshefte für Mathematik, 183(3), 437–463, 2017.MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Burger, M. Di  Francesco, J.-F. Pietschmann, B. Schlake Nonlinear cross-diffusion with size exclusion, SIAM Journal of Mathematical Analysis, 42(6):2842–2871, 2010.MathSciNetzbMATHGoogle Scholar
  5. 5.
    M. Burger, S. Hittmeir, H. Ranetbauer, M.-T. Wolfram Lane formation by side-stepping, SIAM Journal on Mathematical Analysis, 48(2):981–1005, 2016.MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Bruna, S. J. Chapman Diffusion of multiple species with excluded-volume effects, The Journal of chemical physics, 137(20):204116, 2012.Google Scholar
  7. 7.
    M. Bruna, S. J. Chapman Excluded-volume effects in the diffusion of hard spheres, Physical Review E, 85(1):011103, 2012.Google Scholar
  8. 8.
    L. Chen, A. Jüngel Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM journal on mathematical analysis, 36(1):301–322, 2004.MathSciNetzbMATHGoogle Scholar
  9. 9.
    L. Chen, A. Jüngel Analysis of a parabolic cross-diffusion population model without self-diffusion, Journal of Differential Equations, 224(1):39–59, 2006.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Y. Chen, T. Kolokolnikov A minimal model of predator swarm interactions, Journal of The Royal Society Interface, 11(94):20131208, 2014.Google Scholar
  11. 11.
    X. Chen and A. Jüngel Weak-strong uniqueness of renormalized solutions to reaction-cross-diffusion systems, Submitted for publication, 2018, http://www.asc.tuwien.ac.at/~juengel/publications/pdf/p18xchen.pdf
  12. 12.
    V. Ehrlacher, A. Bakhta Cross-Diffusion systems with non-zero flux and moving boundary conditions, ESAIM: Mathematical Modelling and Numerical Analysis, 2017.Google Scholar
  13. 13.
    L.C. Evans Partial Differential Equations, American Mathematical Society, 2010.Google Scholar
  14. 14.
    J. A. Griepentrog On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach Center Publications, WIAS preprint 898, 66:153–164, 2004.Google Scholar
  15. 15.
    J. A. Griepentrog, L. Recke Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems, Journal of Evolution Equations, 10(2):341–375, 2010.MathSciNetzbMATHGoogle Scholar
  16. 16.
    A. Jüngel, I. V. Stelzer Existence Analysis of Maxwell-Stefan Systems for Multicomponent Mixtures, SIAM Journal of Mathematical Analysis, 45(4):2421–2440, 2012.MathSciNetzbMATHGoogle Scholar
  17. 17.
    A. Jüngel, N. Zamponi Analysis of degenerate Cross-Diffusion population models with volume filling, Annales de l’Institut Henri Poincare (C) Nonlinear Analysis, Elsevier, 2015.Google Scholar
  18. 18.
    A. Jüngel, X. Chen A note on the uniqueness of weak solutions to a class of cross-diffusion systems, arXiv:1706.08812, 2017.
  19. 19.
    A. Jüngel The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28(6):1963, 2015.MathSciNetzbMATHGoogle Scholar
  20. 20.
    K. H. W. Küfner Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model, Nonlinear Differential Equations and Applications NoDEA, 3(4):421–444, 1996.MathSciNetzbMATHGoogle Scholar
  21. 21.
    G. Leoni, G. Rolland A First Course in Sobolev Spaces, AMS Graduate studies in mathematics, 2009.Google Scholar
  22. 22.
    T. Lepoutre, M. Pierre, G. Rolland Global well-posedness of a conservative relaxed cross diffusion system, SIAM Journal on Mathematical Analysis, 44(3):1674–1693, 2012.MathSciNetzbMATHGoogle Scholar
  23. 23.
    D. Le,T. T. Nguyen Everywhere regularity of solutions to a class of strongly coupled degenerate parabolic systems. Communications in Partial Differential Equations, 31(2):307–324, 2006.MathSciNetzbMATHGoogle Scholar
  24. 24.
    J. L. Lions Quelques méthodes de résolution des problémes aux limites non linéaires. Etudes mathématiques, Dunod, 1969.Google Scholar
  25. 25.
    K. J. Painter Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bulletin of Mathematical Biology, 71(5):1117–1147, 2009.MathSciNetzbMATHGoogle Scholar
  26. 26.
    K. J. Painter, T. Hillen Volume-filling and quorum-sensing in models for chemosensitive movement Can. Appl. Math. Quart, 10(4):501–543, 2002.MathSciNetzbMATHGoogle Scholar
  27. 27.
    B. Schlake Mathematical Models for Particle Transport: Crowded Motion, PhD thesis, WWU Münster, 2011.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany
  2. 2.Department MathematikFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany
  3. 3.CERMICSÉcole des Ponts ParisTechMarne-La-Vallée Cedex 2France
  4. 4.Inria Paris, MATHERIALS Project-TeamParis Cedex 12France

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