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January  2017, 37(1): 131-167. doi: 10.3934/dcds.2017006

On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations

Ciprian G. Gal

Department of Mathematics, Florida International University, Miami, FL 33199, USA

Received  February 2016 Revised  September 2016 Published  November 2016

We consider a doubly nonlocal Cahn-Hilliard equation for the nonlocal phase-separation of a two-component material in a bounded domain in the case when mass transport exhibits non-Fickian behavior. Such equations are important for phase-segregation phenomena that exhibit non-standard (anomalous) behaviors. Recently, four different cases were proposed to handle this important equation and the two levels of nonlocality and interaction that are present in the equation. The so-called strong-to-weak interaction case (when one kernel is integrable in some sense while the other is not) was investigated recently for the doubly nonlocal parabolic equation with a regular polynomial potential. In this contribution, we address the so-called strong-to-strong interaction case when both kernels are strongly singular and non-integrable in a suitable sense. We establish well-posedness results along with some regularity and long-time results in terms of finite dimensional global attractors.

Keywords: Doubly nonlocal Cahn-Hilliard equations, strong-to-strong interaction, existence and regularity of solutions, attractors, phase segregation.
Mathematics Subject Classification: 35R09, 37L30, 82C2.
Citation: Ciprian G. Gal. On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 131-167. doi: 10.3934/dcds.2017006
References:
[1]

H. Abels, S. Bosia and M. Grasselli, Cahn-Hilliard equation with nonlocal singular free energies, Ann. Mat. Pura Appl., 194 (2015), 1071-1106.  doi: 10.1007/s10231-014-0411-9.  Google Scholar

[2]

P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277.  doi: 10.1016/j.jde.2004.07.003.  Google Scholar

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[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.  Google Scholar

[5] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158.  Google Scholar
[6]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition, De Gruyter Studies in Mathematics 19, Berlin, 2011.  Google Scholar

[7]

C. G. Gal, Nonlocal Cahn-Hilliard equations with fractional dynamic boundary conditions, to appear. Google Scholar

[8]

C. G. Gal, Doubly Nonlocal Cahn-Hilliard Equations, submitted. Google Scholar

[9]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-13.  doi: 10.1016/S0022-247X(02)00425-0.  Google Scholar

[10]

C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179.  doi: 10.3934/dcds.2014.34.145.  Google Scholar

[11]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions, Ⅰ. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61.  doi: 10.1007/BF02181479.  Google Scholar

[12]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

[13]

C. G. Gal and M. Warma, Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions, Evolution Equations and Control Theory, 5 (2016), 61-103.  doi: 10.3934/eect.2016.5.61.  Google Scholar

[14]

Q. Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.  Google Scholar

[15]

S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.  doi: 10.3934/dcdss.2011.4.653.  Google Scholar

[16]

S.-O. Londen and H. Petzeltová, Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735.  doi: 10.1016/j.jmaa.2011.02.003.  Google Scholar

[17]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 724-735.  doi: 10.1007/BF02413623.  Google Scholar

[18]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: Evolutionary equations. Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, Ⅳ (2008), 103-200 doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[19]

A. Novick-Cohen, The Cahn-Hilliard equation, Evolutionary equations, Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, Ⅳ (2008), 201-228 doi: 10.1016/S1874-5717(08)00004-2.  Google Scholar

[20]

S. P. Neumana and D. M. Tartakovsky, Perspective on theories of non-Fickian transport in heterogeneous media, Advances in Water Resources, 32 (2009), 670-680.  doi: 10.1016/j.advwatres.2008.08.005.  Google Scholar

[21]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, in Order and Chaos, Vol. 10 (ed. T. Bountis), Patras University Press, 2008. Google Scholar

[22]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4.  Google Scholar

[23]

E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ/B: Nonlinear monotone operators, Springer-Verlag, New York, 1990. Google Scholar

[24]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

H. Abels, S. Bosia and M. Grasselli, Cahn-Hilliard equation with nonlocal singular free energies, Ann. Mat. Pura Appl., 194 (2015), 1071-1106.  doi: 10.1007/s10231-014-0411-9.  Google Scholar

[2]

P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277.  doi: 10.1016/j.jde.2004.07.003.  Google Scholar

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[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.  Google Scholar

[5] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158.  Google Scholar
[6]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition, De Gruyter Studies in Mathematics 19, Berlin, 2011.  Google Scholar

[7]

C. G. Gal, Nonlocal Cahn-Hilliard equations with fractional dynamic boundary conditions, to appear. Google Scholar

[8]

C. G. Gal, Doubly Nonlocal Cahn-Hilliard Equations, submitted. Google Scholar

[9]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-13.  doi: 10.1016/S0022-247X(02)00425-0.  Google Scholar

[10]

C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179.  doi: 10.3934/dcds.2014.34.145.  Google Scholar

[11]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions, Ⅰ. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61.  doi: 10.1007/BF02181479.  Google Scholar

[12]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

[13]

C. G. Gal and M. Warma, Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions, Evolution Equations and Control Theory, 5 (2016), 61-103.  doi: 10.3934/eect.2016.5.61.  Google Scholar

[14]

Q. Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.  Google Scholar

[15]

S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.  doi: 10.3934/dcdss.2011.4.653.  Google Scholar

[16]

S.-O. Londen and H. Petzeltová, Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735.  doi: 10.1016/j.jmaa.2011.02.003.  Google Scholar

[17]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 724-735.  doi: 10.1007/BF02413623.  Google Scholar

[18]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: Evolutionary equations. Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, Ⅳ (2008), 103-200 doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[19]

A. Novick-Cohen, The Cahn-Hilliard equation, Evolutionary equations, Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, Ⅳ (2008), 201-228 doi: 10.1016/S1874-5717(08)00004-2.  Google Scholar

[20]

S. P. Neumana and D. M. Tartakovsky, Perspective on theories of non-Fickian transport in heterogeneous media, Advances in Water Resources, 32 (2009), 670-680.  doi: 10.1016/j.advwatres.2008.08.005.  Google Scholar

[21]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, in Order and Chaos, Vol. 10 (ed. T. Bountis), Patras University Press, 2008. Google Scholar

[22]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4.  Google Scholar

[23]

E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ/B: Nonlinear monotone operators, Springer-Verlag, New York, 1990. Google Scholar

[24]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

Table 1.  $X\subset \mathbb{R}^{n}$ is a bounded domain with Lipschitz continuous boundary $\partial X.$ The general model covered is the mass-conserved one given by (1.5) with the following choices of operators $A,B$. A physically relevant choice that satisfies our assumptions is the double-well potential $F\left( s\right) =\theta s^{4}-\theta _{c}s^{2}$, $0<\theta <\theta _{c}.$
Model Classical CHE Doubly nonlocal CHE, case (2)
$ \mathit{A}$ $ -\Delta _{X,N}$ $ (-\Delta )_{X,N}^{s_{1}}, s_{1}\in \left( 1/2,1\right)$
$ \mathit{B}$ $ -\Delta _{X,N}$ $ (-\Delta )_{X,N}^{s_{2}}, s_{2}\in \left( 1/2,1\right)$
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Model Classical CHE Doubly nonlocal CHE, case (2)
$ \mathit{A}$ $ -\Delta _{X,N}$ $ (-\Delta )_{X,N}^{s_{1}}, s_{1}\in \left( 1/2,1\right)$
$ \mathit{B}$ $ -\Delta _{X,N}$ $ (-\Delta )_{X,N}^{s_{2}}, s_{2}\in \left( 1/2,1\right)$
Table 2.  The information is the same as in Table 1.
Model CHE: anamolous transport CHE: nonlocal strong energy
$ \mathit{A}$ $ (-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$ $-\Delta _{X,N}$
$ \mathit{B}$ $ -\Delta _{X,N}$ $ (-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$
Table Options

Download as excel

Model CHE: anamolous transport CHE: nonlocal strong energy
$ \mathit{A}$ $ (-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$ $-\Delta _{X,N}$
$ \mathit{B}$ $ -\Delta _{X,N}$ $ (-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$
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