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

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

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.

Citation: Ciprian G. Gal. On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 131-167. doi: 10.3934/dcds.2017006
References:
[1]

H. AbelsS. 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. 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

[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, (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, (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. AbelsS. 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. 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

[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, (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, (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)$
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)$
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)$
[1]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630

[2]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145

[3]

Francesco Della Porta, Maurizio Grasselli. Convective nonlocal Cahn-Hilliard equations with reaction terms. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1529-1553. doi: 10.3934/dcdsb.2015.20.1529

[4]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741

[5]

Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052

[6]

Elena Bonetti, Pierluigi Colli, Luca Scarpa, Giuseppe Tomassetti. A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1001-1022. doi: 10.3934/cpaa.2018049

[7]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027

[8]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033

[9]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113

[10]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[11]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163

[12]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423

[13]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73

[14]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127

[15]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669

[16]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308

[17]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31

[18]

Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125

[19]

T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491

[20]

Irena Pawłow, Wojciech M. Zajączkowski. On a class of sixth order viscous Cahn-Hilliard type equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 517-546. doi: 10.3934/dcdss.2013.6.517

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (70)
  • HTML views (3)
  • Cited by (1)

Other articles
by authors

[Back to Top]