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On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations
Department of Mathematics, Florida International University, Miami, FL 33199, USA |
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.
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. |
Model | Classical CHE | Doubly nonlocal CHE, case (2) |
Model | Classical CHE | Doubly nonlocal CHE, case (2) |
Model | CHE: anamolous transport | CHE: nonlocal strong energy |
Model | CHE: anamolous transport | CHE: nonlocal strong energy |
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