\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the Logarithmic Cahn-Hilliard Equation with General Proliferation Term

  • *Corresponding author: Rim Mheich

    *Corresponding author: Rim Mheich 
Abstract Full Text(HTML) Figure(11) Related Papers Cited by
  • Our aim in this article is to study the well-posedness of the generalized logarithmic nonlinear Cahn-Hilliard equation with regularization and proliferation terms. We are interested in studying the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem and majorate the rate of convergence between the solutions of the Cahn-Hilliard equation and the regularized one. Additionally, we present some further regularity results and subsequently prove a strict separation property of the solution. Finally, we provide some numerical simulations to compare the solution with and without the regularization term, and more.

    Mathematics Subject Classification: Primary: 35B40, 35B41, 35B45; Secondary: 35K55.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The solution $ u $ for $ \varepsilon = 0 $

    Figure 2.  The solution $ u $ for $ \varepsilon = 0.01 $.

    Figure 3.  The solution $ u $ for $ \varepsilon = 0.1 $.

    Figure 4.  The solution $ u $ for $ \varepsilon = 1 $.

    Figure 5.  The solution $ u $ for $ \lambda = 0 $

    Figure 6.  The solution $ u $ for $ \lambda = 5 $

    Figure 7.  The solution $ u $ for $ \lambda = 10 $

    Figure 8.  The solution $ u $ for $ \lambda = 15 $

    Figure 9.  The solution $ u $ for $ \lambda = 0 $

    Figure 10.  The solution $ u $ for $ \lambda = 5 $

    Figure 11.  The solution $ u $ for $ \lambda = 10\;at\;t=0.65 $

  • [1] J. W. Cahn and J. W. Hilliard, Free energy of a nonuniform systemI. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. 
    [2] J. W. Cahn, On spinodal decomposition, Acta Metall, 9 (1961), 795-801. 
    [3] V. Chalupecki, Numerical studies of Cahn-Hilliard equations and applications in image processing, Proceedings of Czech-Japanese Seminar in Applied Mathematics, 4-7 August, 2004, Czech Technical University in Prague, 2004.
    [4] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79, (2011), 561-596. doi: 10.1007/s00032-011-0165-4.
    [5] L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Cont. Dyn. Syst. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.
    [6] D. Cohen and J. M. Murray, A generalized diffusion model for growth and dispersion in a population, J. Math. Biol., 12 (1981), 237-249.  doi: 10.1007/BF00276132.
    [7] I. C. DolcettaS. F. Vita and R. March, Area-preserving curve-shortening flows: From phase separation to image processing, Interfaces Free Bound., 4 (2002), 325-343.  doi: 10.4171/IFB/64.
    [8] M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singular perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.
    [9] C. M. Elliott, The Cahn-Hilliard model for the kinetics of phases separation, in Mathematical Models for Phase Change Problems, Rodrigues, J.F. (ed.), International Series of Numerical Mathematics, vol. 88. Birkhöuser, Basel (1989).
    [10] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stockes systems with singular potentials, Dyn. PDF, 9 (2012), 273-304.  doi: 10.4310/DPDE.2012.v9.n4.a1.
    [11] E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129, 7 pp.
    [12] I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E, 74 (2006), 0319021, 8 pp. doi: 10.1103/PhysRevE.74.031902.
    [13] R. V. Kohn and F. Otto, Upper bounds for coarsening rates, Commun. Math. Phys., 229 (2002), 375-395.  doi: 10.1007/s00220-002-0693-4.
    [14] J. S. Langer, Theory of spinodal decomposition in alloys, Ann. Phys., 65 (1975), 53-86. 
    [15] S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part Ⅰ: Probability and wavelength estimate, Commun. Math. Phys., 195 (1998), 435-464.  doi: 10.1007/s002200050397.
    [16] S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: nonlinear dynamics, Arch. Ration. Mech. Anal., 151 (2000), 187-219.  doi: 10.1007/s002050050196.
    [17] R. Mheich, Cahn-Hilliard Equation with Regularization Term, Asymp. Anal., 133 (2023), 499-533.  doi: 10.3233/ASY-221821.
    [18] M. K. Miller, J. M. Hyde and M. G. Hetherington, et al., Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models-Ⅰ. Introduction and methodolog, Acta Metall. Mater., 43 (1995), 3385-3401.
    [19] M. K. Miller, J. M. Hyde and M. G. Hetherington, et al., Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models-Ⅱ. Development of domain size and composition amplitude, Acta Metall. Mater., 43 (1995), 3403-3413.
    [20] M. K. Miller, J. M. Hyde and M. G. Hetherington, et al., Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models-Ⅲ. Development of morphology, Acta Metall. Mater., 43 (1995), 3415-3426.
    [21] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potential, Math. Meth. Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.
    [22] A. Miranville, Asymptotic behavior of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1308-1321.  doi: 10.1080/00036811.2012.671301.
    [23] A. Miranville, A generalized Cahn-Hilliard equation with logarithmic potentials, in Continuous and Distribued Systems II, Springer, 137-148, 2015. doi: 10.1007/978-3-319-19075-4_8.
    [24] A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 2019. doi: 10.1137/1.9781611975925.
    [25] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985. 
    [26] A. Novick-Cohen, The Cahn-Hilliard equation, In: Dafermos, C.M., Pokorny, M. (eds.) Handbook of Differential Equations, Evolutionary Partial Differential Equations, vol. 4, pp. 201-228. Elsevier, Amsterdam (2008). doi: 10.1016/S1874-5717(08)00004-2.
    [27] Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.
    [28] A. OronS. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980. 
    [29] U. Thiele and E. Knobloch, Thin liquid films on slightly inclined heated plate, Phys. D, 190 (2004), 213-248.  doi: 10.1016/j.physd.2003.09.048.
    [30] S. Tremaine, On the origin of irregular structure in Saturn's ring, Astron. J., 125 (2003), 894-901. 
    [31] S. Villain-Guillot, Phases modulées et dynamique de Cahn-Hilliard, Université Bordeaux 1, 2010.
  • 加载中

Figures(11)

SHARE

Article Metrics

HTML views(248) PDF downloads(121) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return