Global well-posedness in  uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

Sergey Zelik; Jon Pennant

doi:10.3934/cpaa.2013.12.461

Previous Article
The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response
CPAA Home
This Issue
Next Article
Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros

January 2013, 12(1): 461-480. doi: 10.3934/cpaa.2013.12.461

Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

Sergey Zelik ^1, and Jon Pennant ^2,

1.	Department of Mathematics, University of Surrey, Guildford, GU2 7XH
2.	University of Surrey, Guildford, GU2 7XH, China

Received July 2011 Revised May 2012 Published September 2012

Abstract
Related Papers

We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.

Keywords: infinite-energy solutions., unbounded domains, Cahn-Hilliard equation.

Mathematics Subject Classification: 35B41, 35L05, 74K1.

Citation: Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461

References:

[1]	F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, J. Differential Equations, 83 (1990), 85. doi: 10.1016/0022-0396(90)90070-6. Google Scholar
[2]	A. Babin, Global attractors in PDE. Handbook of dynamical systems,, Vol. 1B, (2006), 983. doi: 10.1016/S1874-575X(06)80039-1. Google Scholar
[3]	A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, (1992). Google Scholar
[4]	A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain,, Proc. Royal. Soc. Edimburgh, 116A (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar
[5]	A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain,, Quarterly of Applied Mathematics, 64 (2006), 94. Google Scholar
[6]	J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts,, Comm. Pure Appl. Math., 52 (1999), 839. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. Google Scholar
[7]	J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar
[8]	L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation,, rch. Rational Mech. Anal., 133 (1995), 129. doi: 10.1007/BF00376814. Google Scholar
[9]	L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4. Google Scholar
[10]	A. Debussche, A singular perturbation of the Cahn-Hilliard equation,, Asymptotic Anal., 4 (1991), 161. doi: 10.3233/ASY-1991-4202. Google Scholar
[11]	T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$,, Journal Diff. Eqns., 252 (2012), 2771. doi: 10.1016/j.jde.2011.08.052. Google Scholar
[12]	A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation,, Comm. Pure Appl. Anal., 6 (2007), 1075. doi: 10.3934/cpaa.2007.6.1075. Google Scholar
[13]	A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, (). Google Scholar
[14]	C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems,, Internat. Ser. Numer. Math., 88 (1989), 35. Google Scholar
[15]	M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain,, Comm. Pure Appl. Math., 54 (2001), 625. doi: 10.1002/cpa.1011. Google Scholar
[16]	M. Efendiev, H. Gajewski and S. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity,, Adv. Differential Equations, 7 (2002), 1073. Google Scholar
[17]	M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, Math. Nachr., 272 (2004), 11. doi: 10.1002/mana.200310186. Google Scholar
[18]	J. Evans, V. Galaktionov and J. Williams, Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity,, SIAM Journal on Mathematical Analysis, 38 (2006), 64. Google Scholar
[19]	M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, Comm. Partial Differential Equations, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar
[20]	M. Grasselli, Maurizio, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term,, J. Evol. Equ., 9 (2009), 371. doi: 10.1007/s00028-009-0017-7. Google Scholar
[21]	M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term,, J. Differential Equations, 239 (2007), 38. doi: 10.1016/j.jde.2007.05.003. Google Scholar
[22]	V. Kalantarov, Global behavior of the solutions of some fourth-order nonlinear equations. (Russian),, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 163 (1987), 66. doi: 10.1007/BF02208712. Google Scholar
[23]	T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts,, Comm. Pure Appl. Math., 58 (2005), 1077. Google Scholar
[24]	A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations,, Hokkaido Math. J., 38 (2009), 315. Google Scholar
[25]	A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar
[26]	A. Miranville and A. S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Methods Appl. Sci., 28 (2005), 709. doi: 10.1002/mma.590. Google Scholar
[27]	A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar
[28]	A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation,, Journal of Applied Analysis and Computation, 1 (2011), 523. Google Scholar
[29]	A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965. Google Scholar
[30]	A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys,, Appl. Anal., 47 (1992), 241. doi: 10.1080/00036819208840143. Google Scholar
[31]	Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases,, Phys. Rev. Letters, 58 (1987), 836. doi: 10.1103/PhysRevLett.58.836. Google Scholar
[32]	E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems,, Phys. D, 192 (2004), 279. doi: 10.1016/j.physd.2004.01.024. Google Scholar
[33]	R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition,, Applied Mathematical Sciences, 68 (1997). Google Scholar
[34]	J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation,, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459. doi: 10.1016/S0294-1449(98)80031-0. Google Scholar
[35]	S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II,, Int. Math. Ser. Springer, 7 (2008), 255. doi: 10.1007/978-0-387-75219-8_6. Google Scholar
[36]	S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip,, Glasg. Math. J., 49 (2007), 525. doi: 10.1017/S0017089507003849. Google Scholar
[37]	S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains,, J. Dynam. Differential Equations, 19 (2007), 1. doi: 10.1007/s10884-006-9007-4. Google Scholar
[38]	S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, Comm. Pure Appl. Math., 56 (2003), 584. doi: 10.1002/cpa.10068. Google Scholar

show all references

References:

[1]	F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, J. Differential Equations, 83 (1990), 85. doi: 10.1016/0022-0396(90)90070-6. Google Scholar
[2]	A. Babin, Global attractors in PDE. Handbook of dynamical systems,, Vol. 1B, (2006), 983. doi: 10.1016/S1874-575X(06)80039-1. Google Scholar
[3]	A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, (1992). Google Scholar
[4]	A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain,, Proc. Royal. Soc. Edimburgh, 116A (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar
[5]	A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain,, Quarterly of Applied Mathematics, 64 (2006), 94. Google Scholar
[6]	J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts,, Comm. Pure Appl. Math., 52 (1999), 839. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. Google Scholar
[7]	J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar
[8]	L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation,, rch. Rational Mech. Anal., 133 (1995), 129. doi: 10.1007/BF00376814. Google Scholar
[9]	L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4. Google Scholar
[10]	A. Debussche, A singular perturbation of the Cahn-Hilliard equation,, Asymptotic Anal., 4 (1991), 161. doi: 10.3233/ASY-1991-4202. Google Scholar
[11]	T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$,, Journal Diff. Eqns., 252 (2012), 2771. doi: 10.1016/j.jde.2011.08.052. Google Scholar
[12]	A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation,, Comm. Pure Appl. Anal., 6 (2007), 1075. doi: 10.3934/cpaa.2007.6.1075. Google Scholar
[13]	A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, (). Google Scholar
[14]	C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems,, Internat. Ser. Numer. Math., 88 (1989), 35. Google Scholar
[15]	M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain,, Comm. Pure Appl. Math., 54 (2001), 625. doi: 10.1002/cpa.1011. Google Scholar
[16]	M. Efendiev, H. Gajewski and S. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity,, Adv. Differential Equations, 7 (2002), 1073. Google Scholar
[17]	M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, Math. Nachr., 272 (2004), 11. doi: 10.1002/mana.200310186. Google Scholar
[18]	J. Evans, V. Galaktionov and J. Williams, Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity,, SIAM Journal on Mathematical Analysis, 38 (2006), 64. Google Scholar
[19]	M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, Comm. Partial Differential Equations, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar
[20]	M. Grasselli, Maurizio, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term,, J. Evol. Equ., 9 (2009), 371. doi: 10.1007/s00028-009-0017-7. Google Scholar
[21]	M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term,, J. Differential Equations, 239 (2007), 38. doi: 10.1016/j.jde.2007.05.003. Google Scholar
[22]	V. Kalantarov, Global behavior of the solutions of some fourth-order nonlinear equations. (Russian),, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 163 (1987), 66. doi: 10.1007/BF02208712. Google Scholar
[23]	T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts,, Comm. Pure Appl. Math., 58 (2005), 1077. Google Scholar
[24]	A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations,, Hokkaido Math. J., 38 (2009), 315. Google Scholar
[25]	A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar
[26]	A. Miranville and A. S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Methods Appl. Sci., 28 (2005), 709. doi: 10.1002/mma.590. Google Scholar
[27]	A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar
[28]	A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation,, Journal of Applied Analysis and Computation, 1 (2011), 523. Google Scholar
[29]	A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965. Google Scholar
[30]	A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys,, Appl. Anal., 47 (1992), 241. doi: 10.1080/00036819208840143. Google Scholar
[31]	Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases,, Phys. Rev. Letters, 58 (1987), 836. doi: 10.1103/PhysRevLett.58.836. Google Scholar
[32]	E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems,, Phys. D, 192 (2004), 279. doi: 10.1016/j.physd.2004.01.024. Google Scholar
[33]	R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition,, Applied Mathematical Sciences, 68 (1997). Google Scholar
[34]	J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation,, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459. doi: 10.1016/S0294-1449(98)80031-0. Google Scholar
[35]	S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II,, Int. Math. Ser. Springer, 7 (2008), 255. doi: 10.1007/978-0-387-75219-8_6. Google Scholar
[36]	S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip,, Glasg. Math. J., 49 (2007), 525. doi: 10.1017/S0017089507003849. Google Scholar
[37]	S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains,, J. Dynam. Differential Equations, 19 (2007), 1. doi: 10.1007/s10884-006-9007-4. Google Scholar
[38]	S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, Comm. Pure Appl. Math., 56 (2003), 584. doi: 10.1002/cpa.10068. Google Scholar

[1]	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
[2]	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
[3]	Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361
[4]	Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[5]	Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[6]	Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630
[7]	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
[8]	Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 609-629. doi: 10.3934/dcds.2007.19.609
[9]	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
[10]	Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[11]	Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[12]	S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[13]	Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[14]	Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[15]	Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[16]	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
[17]	Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030
[18]	Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[19]	Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[20]	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

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences

Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors