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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
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