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

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

Mathias Wilke 1,

1. 

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

Received  March 2012 Revised  June 2012 Published  October 2012

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 "Function Spaces, Differential Operators and Nonlinear Analysis" (eds. H.-J. Schmeisser and H. Triebel), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9-126.  Google Scholar

[2]

H. Amann, "Linear and Quasilinear Parabolic Problems," Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995.  Google Scholar

[3]

A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania 2000), 47 (2001), 3455-3466.  Google Scholar

[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), 1448-1462. doi: 10.1002/mana.200410431.  Google Scholar

[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), viii+114.  Google Scholar

[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), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[7]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.  Google Scholar

[8]

M. Girardi and L. Weis, Criteria for R-boundedness of operator families, Evolution equations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 234 (2003), 203-221.  Google Scholar

[9]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[10]

N. J. Kalton and L. Weis, The $H^ \infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345. doi: 10.1007/s002080100231.  Google Scholar

[11]

M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Equ., 10 (2010), 443-463. doi: 10.1007/s00028-010-0056-0.  Google Scholar

[12]

A. Miranville, Existence of solutions for a Cahn-Hilliard-Gurtin model, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 845-850. doi: 10.1016/S0764-4442(00)01731-6.  Google Scholar

[13]

A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., (2003), 165-185.  Google Scholar

[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), 315-345.  Google Scholar

[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), 244-268. doi: 10.1007/s00033-005-0017-6.  Google Scholar

[16]

A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360.  Google Scholar

[17]

A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Phys. D, 158 (2001), 233-257. doi: 10.1016/S0167-2789(01)00317-7.  Google Scholar

[18]

A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation, Nonlinear Anal. Real World Appl., 7 (2006), 285-307.  Google Scholar

[19]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.  Google Scholar

[20]

J. Prüss and M. Wilke, On conserved Penrose-Fife type systems, Parabolic Problems, The Herbert Amann Festschrift, Progress in nonlinear differential equations and their applications, Birkhäuser, Basel, 80 (2011), 541-576. Google Scholar

[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, I. Studia Math., 44 (1972), 47-60.  Google Scholar

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (eds. H.-J. Schmeisser and H. Triebel), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9-126.  Google Scholar

[2]

H. Amann, "Linear and Quasilinear Parabolic Problems," Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995.  Google Scholar

[3]

A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania 2000), 47 (2001), 3455-3466.  Google Scholar

[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), 1448-1462. doi: 10.1002/mana.200410431.  Google Scholar

[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), viii+114.  Google Scholar

[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), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[7]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.  Google Scholar

[8]

M. Girardi and L. Weis, Criteria for R-boundedness of operator families, Evolution equations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 234 (2003), 203-221.  Google Scholar

[9]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[10]

N. J. Kalton and L. Weis, The $H^ \infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345. doi: 10.1007/s002080100231.  Google Scholar

[11]

M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Equ., 10 (2010), 443-463. doi: 10.1007/s00028-010-0056-0.  Google Scholar

[12]

A. Miranville, Existence of solutions for a Cahn-Hilliard-Gurtin model, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 845-850. doi: 10.1016/S0764-4442(00)01731-6.  Google Scholar

[13]

A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., (2003), 165-185.  Google Scholar

[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), 315-345.  Google Scholar

[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), 244-268. doi: 10.1007/s00033-005-0017-6.  Google Scholar

[16]

A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360.  Google Scholar

[17]

A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Phys. D, 158 (2001), 233-257. doi: 10.1016/S0167-2789(01)00317-7.  Google Scholar

[18]

A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation, Nonlinear Anal. Real World Appl., 7 (2006), 285-307.  Google Scholar

[19]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.  Google Scholar

[20]

J. Prüss and M. Wilke, On conserved Penrose-Fife type systems, Parabolic Problems, The Herbert Amann Festschrift, Progress in nonlinear differential equations and their applications, Birkhäuser, Basel, 80 (2011), 541-576. Google Scholar

[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, I. Studia Math., 44 (1972), 47-60.  Google Scholar

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