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Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system (2307.05985v2)

Published 12 Jul 2023 in math.AP, cs.NA, and math.NA

Abstract: We study some properties of a multi-species degenerate Ginzburg-Landau energy and its relation to a cross-diffusion Cahn-Hilliard system. The model is motivated by multicomponent mixtures where crossdiffusion effects between the different species are taken into account, and where only one species does separate from the others. Using a comparison argument, we obtain strict bounds on the minimizers from which we can derive first-order optimality conditions, revealing a link with the single-species energy, and providing enough regularity to qualify the minimizers as stationary solutions of the evolution system. We also discuss convexity properties of the energy as well as long time asymptotics of the time-dependent problem. Lastly, we introduce a structure-preserving finite volume scheme for the time-dependent problem and present several numerical experiments in one and two spatial dimensions.

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References (56)
  1. H. Abels and M. Wilke. Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy. Nonlinear Analysis: Theory, Methods & Applications, 67(11):3176–3193, Dec. 2007.
  2. Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation, May 2021.
  3. M. Bessemoulin-Chatard and F. Filbet. A finite volume scheme for nonlinear degenerate parabolic equations. SIAM Journal on Scientific Computing, 34(5):B559–B583, Jan. 2012.
  4. F. Boyer and S. Minjeaud. Hierarchy of consistent n-component Cahn-Hilliard systems. Math. Mod. Meth. Appl. S., 24(14):2885–2928, 2014.
  5. Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 15(8):1987, 2022.
  6. On existence, uniqueness and stability of solutions to Cahn-Hilliard/Allen-Cahn systems with cross-kinetic coupling, 2022.
  7. Lane formation by side-stepping. SIAM Journal on Mathematical Analysis, 48(2):981–1005, 2016.
  8. L. Caffarelli and N. Muler. An L∞superscript𝐿{L}^{\infty}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal., 133(2):129–144, Dec 1995.
  9. J. Cahn and J. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28(2):258–267, 1958.
  10. Structure preserving finite volume approximation of cross-diffusion systems coupled by a free interface. In Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems, Springer Proceedings in Mathematics & Statistics, pages 205–213, 2023.
  11. Finite-volume scheme for a degenerate cross-diffusion model motivated from ion transport. Numerical Methods for Partial Differential Equations, 35(2):545–575, Mar. 2019.
  12. Finite volumes for the Stefan-Maxwell cross-diffusion system. arXiv:2007.09951 [cs, math], 2020.
  13. C. Cancès and B. Gaudeul. A convergent entropy diminishing finite volume scheme for a cross-diffusion system. SIAM Journal on Numerical Analysis, 58(5):2684–2710, Jan. 2020.
  14. C. Cancès and C. Guichard. Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure. Foundations of Computational Mathematics, 17(6):1525–1584, Dec. 2017.
  15. C. Cancès and A. Zurek. A convergent finite volume scheme for dissipation driven models with volume filling constraint. Numerische Mathematik, 2022.
  16. Convergence of a finite volume scheme for a system of interacting species with cross-diffusion. Numerische Mathematik, 145(3):473–511, July 2020.
  17. C. Chainais-Hillairet. Entropy method and asymptotic behaviours of finite volume schemes. In J. Fuhrmann, M. Ohlberger, and C. Rohde, editors, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, Springer Proceedings in Mathematics & Statistics, pages 17–35, Cham, 2014. Springer International Publishing.
  18. L. Chen and A. Jüngel. Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal., 36(1):301–322 (electronic), 2004.
  19. L. Chen and A. Jüngel. Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Differential Equations, 224(1):39–59, 2006.
  20. Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential. Journal of Computational Physics: X, 3:100031, June 2019.
  21. Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms. IMA Journal of Numerical Analysis, 41(2):935–973, 2021.
  22. L. Desvillettes and K. Fellner. Exponential convergence to equilibrium for nonlinear reaction-diffusion systems arising in reversible chemistry. In C. Pötzsche, C. Heuberger, B. Kaltenbacher, and F. Rendl, editors, System Modeling and Optimization, pages 96–104, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg.
  23. Existence of weak solutions to a Cross-Diffusion Cahn-Hilliard type system. Journal of Differential Equations, 286:578–623, 2021.
  24. C. Elliott and H. Garcke. On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal., 27(2):404–423, 1996.
  25. C. Elliott and H. Garcke. Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Phys. D, 109(3):242–256, 1997.
  26. C. Elliott and S. Luckhaus. A generalized equation for phase separation of a multi-component mixture with interfacial free energy. preprint SFB, 256:195, 1991.
  27. C. Elliott and Z. Songmu. On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal., 96(4):339–357, Dec 1986.
  28. Finite volume methods. Handbook of numerical analysis, 7:713–1018, 2000.
  29. D. Eyre. Systems of Cahn–Hilliard equations. SIAM Journal on Applied Mathematics, 53(6):1686–1712, 1993.
  30. D. J. Eyre. Unconditionally gradient stable time marching the Cahn-Hilliard equation. In Materials Research Society Symposium Proceedings, volume 529, pages 39–46. Materials Research Society, 1998.
  31. Existence and properties of certain critical points of the Cahn-Hilliard energy. Indiana Univ. Math. J., 66:1827–1877, 2017.
  32. M. Gelantalis and M. Westdickenberg. Energy barrier and ΓΓ\Gammaroman_Γ-convergence in the d𝑑ditalic_d-dimensional Cahn–Hilliard equation. Calculus of Variations and Partial Differential Equations, 54(1):791–829, Dec. 2014.
  33. M. Herda and A. Zurek. Study of a structure preserving finite volume scheme for a nonlocal cross-diffusion system, July 2022.
  34. Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, June 2023.
  35. A. Jüngel. The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity, 28(6):1963, 2015.
  36. A. Jüngel. Entropy methods for diffusive partial differential equations, volume 804. Springer, 2016.
  37. A convergent finite-volume scheme for nonlocal cross-diffusion systems for multi-species populations, Feb. 2023.
  38. A. Jüngel and A. Zurek. A finite-volume scheme for a cross-diffusion model arising from interacting many-particle population systems. In R. Klöfkorn, E. Keilegavlen, F. A. Radu, and J. Fuhrmann, editors, Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, volume 323, pages 223–231. Springer International Publishing, Cham, 2020.
  39. A. Jüngel and A. Zurek. A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems. IMA Journal of Numerical Analysis, 43(1):560–589, 2023.
  40. T. Klinkert. Comprehension and optimisation of the co-evaporation deposition of Cu (In, Ga) Se2 absorber layers for very high efficiency thin film solar cells. PhD thesis, Université Pierre et Marie Curie-Paris VI, 2015.
  41. K. Küfner. Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model. NoDEA-Nonlinear Diff., 3:421–444, 1996.
  42. Global well-posedness of a conservative relaxed cross diffusion system. SIAM J. Math. Anal., 44(3):1674–1693, 2012.
  43. Nucleation rate calculation for the phase transition of diblock copolymers under stochastic Cahn–Hilliard dynamics. Multiscale Modeling & Simulation, 11(1):385–409, 2013.
  44. Growth and investigations on the nucleation kinetics of zinc succinate NLO single crystals. International Journal of ChemTech Research, 3:1070–1074, 2011.
  45. A. Miranville. The Cahn-Hilliard Equation: Recent Advances and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2019.
  46. A. Novick-Cohen. The Cahn-Hilliard equation. Handbook of differential equations: evolutionary equations, 4:201–228, 2008.
  47. Optimal l1superscript𝑙1l^{1}italic_l start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-type relaxation rates for the Cahn–Hilliard equation on the line. SIAM Journal on Mathematical Analysis, 51(6):4645–4682, 2019.
  48. F. Otto and M. Westdickenberg. Relaxation to equilibrium in the one-dimensional Cahn–Hilliard equation. SIAM Journal on Mathematical Analysis, 46(1):720–756, 2014.
  49. K. 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.
  50. K. Painter and T. Hillen. Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q., 10(4):501 – 543, 2002.
  51. G. Schimperna. Global attractors for Cahn–Hilliard equations with nonconstant mobility. Nonlinearity, 20(10):2365, Sept. 2007.
  52. G. Schimperna and S. Zelik. Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations. Transactions of the American Mathematical Society, 365(7):3799–3829, July 2013.
  53. G. Troianiello. Elliptic differential equations and obstacle problems. Springer Science & Business Media, 2013.
  54. Nickel-enhanced graphitic ordering of carbon ad-atoms during physical vapor deposition. Carbon, 100:656 – 663, 2016.
  55. S. Wu and J. Xu. Multiphase Allen-Cahn and Cahn-Hilliard models and their discretizations with the effect of pairwise surface tensions. J. Comput. Phys., 343:10–32, 2017.
  56. A. Zurek and A. Jüngel. A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system, Nov. 2020.
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