# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158. [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. [7] C. G. Gal, Nonlocal Cahn-Hilliard equations with fractional dynamic boundary conditions, to appear. [8] C. G. Gal, Doubly Nonlocal Cahn-Hilliard Equations, submitted. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ/B: Nonlinear monotone operators, Springer-Verlag, New York, 1990. [24] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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. [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. [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. [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] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158. [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. [7] C. G. Gal, Nonlocal Cahn-Hilliard equations with fractional dynamic boundary conditions, to appear. [8] C. G. Gal, Doubly Nonlocal Cahn-Hilliard Equations, submitted. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ/B: Nonlinear monotone operators, Springer-Verlag, New York, 1990. [24] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
$X\subset \mathbb{R}^{n}$ 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) $\mathit{A}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s_{1}}, s_{1}\in \left( 1/2,1\right)$ $\mathit{B}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s_{2}}, s_{2}\in \left( 1/2,1\right)$
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 and Continuous Dynamical Systems, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145 [3] Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741 [4] Francesco Della Porta, Maurizio Grasselli. Convective nonlocal Cahn-Hilliard equations with reaction terms. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1529-1553. doi: 10.3934/dcdsb.2015.20.1529 [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 and 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 and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027 [8] Kelong Cheng, Cheng Wang, Steven M. Wise, Zixia Yuan. Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2211-2229. doi: 10.3934/dcdss.2020186 [9] Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 [10] Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113 [11] Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 [12] Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73 [14] Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163 [15] Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127 [16] Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669 [17] Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308 [18] Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31 [19] Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125 [20] T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491

2020 Impact Factor: 1.392