L_p-theory for a Cahn-Hilliard-Gurtin system

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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

Abstract
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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 and 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.
[2]	H. Amann, "Linear and Quasilinear Parabolic Problems," Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., (2003), 165-185.
[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.
[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.
[16]	A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360.
[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.
[18]	A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation, Nonlinear Anal. Real World Appl., 7 (2006), 285-307.
[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.
[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.
[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.

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.
[2]	H. Amann, "Linear and Quasilinear Parabolic Problems," Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., (2003), 165-185.
[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.
[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.
[16]	A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360.
[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.
[18]	A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation, Nonlinear Anal. Real World Appl., 7 (2006), 285-307.
[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.
[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.
[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.

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