L_p-theory for a Cahn-Hilliard-Gurtin system

Previous Article
Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise
EECT Home
This Issue
Next Article

December 2012, 1(2): 393-429. doi: 10.3934/eect.2012.1.393

$L_p$-theory for a Cahn-Hilliard-Gurtin system

1.	Institut für Mathematik, Martin-Luther Universität Halle-Wittenberg, 06099 Halle, Germany

Received March 2012 Revised June 2012 Published October 2012

Abstract
Related Papers

In this paper we study a generalized Cahn-Hilliard equation which was proposed by Gurtin [9]. We prove the existence and uniqueness of a local-in-time solution for a quasilinear version, that is, if the coefficients depend on the solution and its gradient. Moreover we show that local solutions to the corresponding semilinear problem exist globally as long as the physical potential satisfies certain growth conditions. Finally we study the long-time behaviour of the solutions and show that each solution converges to a equilibrium as time tends to infinity.

Keywords: quasilinear elliptic-parabolic system, Lojasiewicz-Simon inequality., convergence to steady states, Cahn-Hilliard-Gurtin equation, global existence, optimal regularity.

Mathematics Subject Classification: Primary: 35K55; Secondary: 35B38, 35B40, 35B65, 82C2.

Citation: Mathias Wilke. $L_p$-theory for a Cahn-Hilliard-Gurtin system. Evolution Equations & Control Theory, 2012, 1 (2) : 393-429. doi: 10.3934/eect.2012.1.393

References:

[1]	H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9.
[2]	H. Amann, "Linear and Quasilinear Parabolic Problems,", Vol. I, (1995).
[3]	A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations,, Proceedings of the Third World Congress of Nonlinear Analysts, 47 (2001), 3455.
[4]	R. Chill, E. Fašangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions,, Math. Nachr., 279 (2006), 1448. doi: 10.1002/mana.200410431.
[5]	R. Denk, M. Hieber and J. Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003).
[6]	R. Denk, M. Hieber and J. Prüss, Optimal $L_p$-$L_q$-estimates for parabolic boundary value problems with inhomogeneous data,, Math. Z., 257 (2007), 193. doi: 10.1007/s00209-007-0120-9.
[7]	G. Dore and A. Venni, On the closedness of the sum of two closed operators,, Math. Z., 196 (1987), 189. doi: 10.1007/BF01163654.
[8]	M. Girardi and L. Weis, Criteria for R-boundedness of operator families,, Evolution equations, 234 (2003), 203.
[9]	M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Phys. D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5.
[10]	N. J. Kalton and L. Weis, The $H^ \infty$-calculus and sums of closed operators,, Math. Ann., 321 (2001), 319. doi: 10.1007/s002080100231.
[11]	M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces,, J. Evol. Equ., 10 (2010), 443. doi: 10.1007/s00028-010-0056-0.
[12]	A. Miranville, Existence of solutions for a Cahn-Hilliard-Gurtin model,, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 845. doi: 10.1016/S0764-4442(00)01731-6.
[13]	A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance,, J. Appl. Math., (2003), 165.
[14]	A. Miranville, A. Piétrus and J. M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance,, Asymptot. Anal., 16 (1998), 315.
[15]	A. Miranville and A. Rougirel, Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations,, Z. Angew. Math. Phys., 57 (2006), 244. doi: 10.1007/s00033-005-0017-6.
[16]	A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations,, Hokkaido Math. J., 38 (2009), 315.
[17]	A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions,, Phys. D, 158 (2001), 233. doi: 10.1016/S0167-2789(01)00317-7.
[18]	A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation,, Nonlinear Anal. Real World Appl., 7 (2006), 285.
[19]	J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces,, Math. Z., 203 (1990), 429. doi: 10.1007/BF02570748.
[20]	J. Prüss and M. Wilke, On conserved Penrose-Fife type systems,, Parabolic Problems, 80 (2011), 541.
[21]	R. Seeley, Interpolation in $L^p$ with boundary conditions,, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, 44 (1972), 47.

show all references

References:

[1]	H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9.
[2]	H. Amann, "Linear and Quasilinear Parabolic Problems,", Vol. I, (1995).
[3]	A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations,, Proceedings of the Third World Congress of Nonlinear Analysts, 47 (2001), 3455.
[4]	R. Chill, E. Fašangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions,, Math. Nachr., 279 (2006), 1448. doi: 10.1002/mana.200410431.
[5]	R. Denk, M. Hieber and J. Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003).
[6]	R. Denk, M. Hieber and J. Prüss, Optimal $L_p$-$L_q$-estimates for parabolic boundary value problems with inhomogeneous data,, Math. Z., 257 (2007), 193. doi: 10.1007/s00209-007-0120-9.
[7]	G. Dore and A. Venni, On the closedness of the sum of two closed operators,, Math. Z., 196 (1987), 189. doi: 10.1007/BF01163654.
[8]	M. Girardi and L. Weis, Criteria for R-boundedness of operator families,, Evolution equations, 234 (2003), 203.
[9]	M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Phys. D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5.
[10]	N. J. Kalton and L. Weis, The $H^ \infty$-calculus and sums of closed operators,, Math. Ann., 321 (2001), 319. doi: 10.1007/s002080100231.
[11]	M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces,, J. Evol. Equ., 10 (2010), 443. doi: 10.1007/s00028-010-0056-0.
[12]	A. Miranville, Existence of solutions for a Cahn-Hilliard-Gurtin model,, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 845. doi: 10.1016/S0764-4442(00)01731-6.
[13]	A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance,, J. Appl. Math., (2003), 165.
[14]	A. Miranville, A. Piétrus and J. M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance,, Asymptot. Anal., 16 (1998), 315.
[15]	A. Miranville and A. Rougirel, Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations,, Z. Angew. Math. Phys., 57 (2006), 244. doi: 10.1007/s00033-005-0017-6.
[16]	A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations,, Hokkaido Math. J., 38 (2009), 315.
[17]	A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions,, Phys. D, 158 (2001), 233. doi: 10.1016/S0167-2789(01)00317-7.
[18]	A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation,, Nonlinear Anal. Real World Appl., 7 (2006), 285.
[19]	J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces,, Math. Z., 203 (1990), 429. doi: 10.1007/BF02570748.
[20]	J. Prüss and M. Wilke, On conserved Penrose-Fife type systems,, Parabolic Problems, 80 (2011), 541.
[21]	R. Seeley, Interpolation in $L^p$ with boundary conditions,, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, 44 (1972), 47.

[1]	Alain Miranville, Giulio Schimperna. On a doubly nonlinear Cahn-Hilliard-Gurtin system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 675-697. doi: 10.3934/dcdsb.2010.14.675
[2]	Sami Injrou, Morgan Pierre. Stable discretizations of the Cahn-Hilliard-Gurtin equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1065-1080. doi: 10.3934/dcds.2008.22.1065
[3]	Gisèle Ruiz Goldstein, Alain Miranville. A Cahn-Hilliard-Gurtin model with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 387-400. doi: 10.3934/dcdss.2013.6.387
[4]	Michel Pierre, Morgan Pierre. Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5347-5377. doi: 10.3934/dcds.2013.33.5347
[5]	Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 165-176. doi: 10.3934/dcdss.2020009
[6]	Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[7]	Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5189-5202. doi: 10.3934/dcds.2013.33.5189
[8]	Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[9]	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
[10]	J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176
[11]	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
[12]	Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025
[13]	Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117
[14]	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
[15]	Xinlong Feng, Yinnian He. On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5387-5400. doi: 10.3934/dcds.2016037
[16]	Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81
[17]	Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21
[18]	Masahiro Kubo, Noriaki Yamazaki. Elliptic-parabolic variational inequalities with time-dependent constraints. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 335-359. doi: 10.3934/dcds.2007.19.335
[19]	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
[20]	Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777

2018 Impact Factor: 1.048

American Institute of Mathematical Sciences