• Previous Article
    Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line
  • CPAA Home
  • This Issue
  • Next Article
    Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency
February  2021, 20(2): 495-510. doi: 10.3934/cpaa.2020277

On the Cahn-Hilliard equation with mass source for biological applications

1. 

Lebanese International University, School of Arts and Sciences, Department of Mathematics and Physics, Bekaa campus, Lebanon

2. 

Lebanese University, Faculty of Sciences, Department of Mathematics, Houch el Oumara, Zahle, Lebanon

3. 

Politehnica University of Bucharest, Splaiul Independentei 313, 060042, Bucharest, Romania

* Corresponding author

Received  December 2019 Revised  September 2020 Published  December 2020

This article deals with some generalizations of the Cahn–Hilliard equation with mass source endowed with Neumann boundary conditions. This equation has many applications in real life e.g. in biology and image inpainting. The first part of this article, discusses the stationary problem of the Cahn–Hilliard equation with mass source. We prove the existence of a unique solution of the associated stationary problem. Then, in the latter part of this article, we consider the evolution problem of the Cahn–Hilliard equation with mass source. We construct a numerical scheme of the model based on a finite element discretization in space and backward Euler scheme in time. Furthermore, after obtaining some error estimates on the numerical solution, we prove that the semi discrete scheme converges to the continuous problem. In addition, we prove the stability of our scheme which allows us to obtain the convergence of the fully discrete problem to the semi discrete one. Finally, we perform the numerical simulations that confirm the theoretical results and demonstrate the performance of our scheme for cancerous tumor growth and image inpainting.

Citation: Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure & Applied Analysis, 2021, 20 (2) : 495-510. doi: 10.3934/cpaa.2020277
References:
[1]

A. C. AristotelousO. A. Karakashian and S. M. Wise, Adaptive, second order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source, IMA J. Numer. Anal, 35 (2015), 1167-1198.  doi: 10.1093/imanum/dru035.  Google Scholar

[2]

A. BertozziS. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Imaging Proc., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.  Google Scholar

[3]

A. BertozziS. Esedoglu and A. Gillette, Analysis of a two-scale Cahn–Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.  doi: 10.1137/060660631.  Google Scholar

[4]

M. BurgerL. He and C. Schönlieb, Cahn–Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 3 (2009), 1129-1167.  doi: 10.1137/080728548.  Google Scholar

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[6]

L. CherfilsH. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation in image inpainting, Inv. Prob. Imaging, 9 (2015), 105-125.  doi: 10.3934/ipi.2015.9.105.  Google Scholar

[7]

L. CherfilsH. Fakih and A. Miranville, On the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation with logarithmic nonlinear terms, SIAM J. Imag. Sci., 8 (2015), 1123-1140.  doi: 10.1137/140985627.  Google Scholar

[8]

L. CherfilsH. Fakih and A. Miranville, A Cahn–Hilliard system with a fidelity term for color image inpainting, J. Math. Imaging Vis., 54 (2016), 117-131.  doi: 10.1007/s10851-015-0593-9.  Google Scholar

[9]

L. CherfilsH. Fakih and A. Miranville, A complex version of the Cahn-Hilliard equation for grayscale image inpainting, J. Multiscale Model. Simul., 15 (2017), 575-605.  doi: 10.1137/15M1040177.  Google Scholar

[10]

L. CherfilsA. 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.  Google Scholar

[11]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn–Hilliard equation with biological applications, Discrete Cont. Dyn.-B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[12]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn–Hilliard equation with dynamic boundary conditions, Discrete Cont. Dyn. S., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.  Google Scholar

[13]

I. C. DolcettaS. F. Vita and R. March, Area-preserving curve-shortening flows: from phase separation to image processing, Interface. Free Bound., 4 (2002), 325-343.  doi: 10.4171/IFB/64.  Google Scholar

[14]

C. M. ElliottD. A. French and F. A. Milner, A second order splitting method for the Cahn–Hilliard equation, Numer. Math., 54 (1989), 575-590.  doi: 10.1007/BF01396363.  Google Scholar

[15]

A. Ern and J. L. Guermond, Elements finis: theorie, applications, mise en oeuvre, Springer-Verlag, Berlin, 2002.  Google Scholar

[16]

H. Fakih, Asymptotic behavior of a generalized Cahn–Hilliard, equation with a mass source, Appl. Anal., 96 (2016), 324-348.  doi: 10.1080/00036811.2015.1135241.  Google Scholar

[17]

H. Fakih, A Cahn–Hilliard equation with a proliferation term for biological and chemical applications, Asympt. Anal., 94 (2015), 71-104.  doi: 10.3233/ASY-151306.  Google Scholar

[18]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[19]

E. Khain and L. M. Sander, A generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 51-129.   Google Scholar

[20]

A. Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comp., 1 (2011), 523-536.   Google Scholar

[21]

A. Miranville, Asymptotic behavior of a generalized Cahn–Hilliard equation with a proliferation, Appl. Anal., 92 (2013), 1308-1321.  doi: 10.1080/00036811.2012.671301.  Google Scholar

[22]

A. Miranville, Existence of solutions to a Cahn–Hilliard type equation with a logarithmic nonlinear term, Mediterr. J. Math., 16 (2019), 1-18.  doi: 10.1007/s00009-018-1284-8.  Google Scholar

[23]

A. Miranville, The Cahn–Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.   Google Scholar

[24]

A. Novick-Cohen and L. A. Segal, Nonlinear Cahn–Hiliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278.   Google Scholar

[25]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.   Google Scholar

[26]

C. B. Schönlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci., 9 (2011), 413-457.   Google Scholar

[27]

S. Villain-Guillot, Phases modulées et dynamique de Cahn–Hilliard, Habilitation thesis, Habilitation thesis, Université Bordeaux 1, 2010. Google Scholar

show all references

References:
[1]

A. C. AristotelousO. A. Karakashian and S. M. Wise, Adaptive, second order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source, IMA J. Numer. Anal, 35 (2015), 1167-1198.  doi: 10.1093/imanum/dru035.  Google Scholar

[2]

A. BertozziS. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Imaging Proc., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.  Google Scholar

[3]

A. BertozziS. Esedoglu and A. Gillette, Analysis of a two-scale Cahn–Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.  doi: 10.1137/060660631.  Google Scholar

[4]

M. BurgerL. He and C. Schönlieb, Cahn–Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 3 (2009), 1129-1167.  doi: 10.1137/080728548.  Google Scholar

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[6]

L. CherfilsH. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation in image inpainting, Inv. Prob. Imaging, 9 (2015), 105-125.  doi: 10.3934/ipi.2015.9.105.  Google Scholar

[7]

L. CherfilsH. Fakih and A. Miranville, On the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation with logarithmic nonlinear terms, SIAM J. Imag. Sci., 8 (2015), 1123-1140.  doi: 10.1137/140985627.  Google Scholar

[8]

L. CherfilsH. Fakih and A. Miranville, A Cahn–Hilliard system with a fidelity term for color image inpainting, J. Math. Imaging Vis., 54 (2016), 117-131.  doi: 10.1007/s10851-015-0593-9.  Google Scholar

[9]

L. CherfilsH. Fakih and A. Miranville, A complex version of the Cahn-Hilliard equation for grayscale image inpainting, J. Multiscale Model. Simul., 15 (2017), 575-605.  doi: 10.1137/15M1040177.  Google Scholar

[10]

L. CherfilsA. 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.  Google Scholar

[11]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn–Hilliard equation with biological applications, Discrete Cont. Dyn.-B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[12]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn–Hilliard equation with dynamic boundary conditions, Discrete Cont. Dyn. S., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.  Google Scholar

[13]

I. C. DolcettaS. F. Vita and R. March, Area-preserving curve-shortening flows: from phase separation to image processing, Interface. Free Bound., 4 (2002), 325-343.  doi: 10.4171/IFB/64.  Google Scholar

[14]

C. M. ElliottD. A. French and F. A. Milner, A second order splitting method for the Cahn–Hilliard equation, Numer. Math., 54 (1989), 575-590.  doi: 10.1007/BF01396363.  Google Scholar

[15]

A. Ern and J. L. Guermond, Elements finis: theorie, applications, mise en oeuvre, Springer-Verlag, Berlin, 2002.  Google Scholar

[16]

H. Fakih, Asymptotic behavior of a generalized Cahn–Hilliard, equation with a mass source, Appl. Anal., 96 (2016), 324-348.  doi: 10.1080/00036811.2015.1135241.  Google Scholar

[17]

H. Fakih, A Cahn–Hilliard equation with a proliferation term for biological and chemical applications, Asympt. Anal., 94 (2015), 71-104.  doi: 10.3233/ASY-151306.  Google Scholar

[18]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[19]

E. Khain and L. M. Sander, A generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 51-129.   Google Scholar

[20]

A. Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comp., 1 (2011), 523-536.   Google Scholar

[21]

A. Miranville, Asymptotic behavior of a generalized Cahn–Hilliard equation with a proliferation, Appl. Anal., 92 (2013), 1308-1321.  doi: 10.1080/00036811.2012.671301.  Google Scholar

[22]

A. Miranville, Existence of solutions to a Cahn–Hilliard type equation with a logarithmic nonlinear term, Mediterr. J. Math., 16 (2019), 1-18.  doi: 10.1007/s00009-018-1284-8.  Google Scholar

[23]

A. Miranville, The Cahn–Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.   Google Scholar

[24]

A. Novick-Cohen and L. A. Segal, Nonlinear Cahn–Hiliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278.   Google Scholar

[25]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.   Google Scholar

[26]

C. B. Schönlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci., 9 (2011), 413-457.   Google Scholar

[27]

S. Villain-Guillot, Phases modulées et dynamique de Cahn–Hilliard, Habilitation thesis, Habilitation thesis, Université Bordeaux 1, 2010. Google Scholar

Figure 1.  (a) Initial datum at $ t = 0 $. (b) Solution after $ 2000 $ iterations. (c) Solution after $ 3000 $ iterations. (d) Solution after $ 4000 $ iterations. (e) Solution after $ 5000 $ iterations. (f) Solution after $ 6000 $ iterations. (g) Solution after $ 8000 $ iterations. (h) Solution after $ 9000 $ iterations
Figure 2.  (a) initial image. (b) mask. (c) Inpainting result
Figure 3.  (a) original image. (b) mask. (c) Inpainting result
[1]

Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001

[2]

Amanda E. Diegel. A C0 interior penalty method for the Cahn-Hilliard equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021030

[3]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393

[4]

Matthias Ebenbeck, Harald Garcke, Robert Nürnberg. Cahn–Hilliard–Brinkman systems for tumour growth. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021034

[5]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[6]

Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934/dcds.2021022

[7]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057

[8]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382

[9]

Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373

[10]

Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227

[11]

Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005

[12]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101

[13]

Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018

[14]

Raghda A. M. Attia, Dumitru Baleanu, Dianchen Lu, Mostafa M. A. Khater, El-Sayed Ahmed. Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021018

[15]

Lu Li. On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021032

[16]

Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021

[17]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[18]

De-han Chen, Daijun jiang. Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021023

[19]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005

[20]

Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3163-3209. doi: 10.3934/dcds.2020402

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (78)
  • HTML views (96)
  • Cited by (0)

Other articles
by authors

[Back to Top]