| Title:
|
On the Caginalp system with dynamic boundary conditions and singular potentials (English) |
| Author:
|
Cherfils, Laurence |
| Author:
|
Miranville, Alain |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
54 |
| Issue:
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2 |
| Year:
|
2009 |
| Pages:
|
89-115 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in $H^2$, the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Łojasiewicz inequality in order to prove the convergence of solutions to steady states. (English) |
| Keyword:
|
Caginalp phase field system |
| Keyword:
|
singular potential |
| Keyword:
|
dynamic boundary conditions |
| Keyword:
|
global existence |
| Keyword:
|
global attractor |
| Keyword:
|
Łojasiewicz-Simon inequality |
| Keyword:
|
convergence to a steady state |
| MSC:
|
35B40 |
| MSC:
|
35B41 |
| MSC:
|
35K55 |
| MSC:
|
35Q53 |
| MSC:
|
80A22 |
| idZBL:
|
Zbl 1212.35012 |
| idMR:
|
MR2491850 |
| DOI:
|
10.1007/s10492-009-0008-6 |
| . |
| Date available:
|
2010-07-20T12:50:47Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140353 |
| . |
| Reference:
|
[1] Abels, H., Wilke, M.: Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy.Nonlinear Anal. 67 (2007), 3176-3193. Zbl 1121.35018, MR 2347608, 10.1016/j.na.2006.10.002 |
| Reference:
|
[2] Aizicovici, S., Feireisl, E.: Long-time stabilization of solutions to a phase-field model with memory.J. Evol. Equ. 1 (2001), 69-84. Zbl 0973.35037, MR 1838321, 10.1007/PL00001365 |
| Reference:
|
[3] Aizicovici, S., Feireisl, E., Issard-Roch, F.: Long-time convergence of solutions to a phase-field system.Math. Methods Appl. Sci. 24 (2001), 277-287. Zbl 0984.35026, MR 1818896, 10.1002/mma.215 |
| Reference:
|
[4] Bates, P. W., Zheng, S.: Inertial manifolds and inertial sets for phase-field equations.J. Dyn. Diff. Equations 4 (1992), 375-398. 10.1007/BF01049391 |
| Reference:
|
[5] Brochet, D., Chen, X., Hilhorst, D.: Finite dimensional exponential attractors for the phase-field model.Appl. Anal. 49 (1993), 197-212. MR 1289743, 10.1080/00036819108840173 |
| Reference:
|
[6] Brokate, M., Sprekels, J.: Hysteresis and phase transitions.Springer New York (1996). Zbl 0951.74002, MR 1411908 |
| Reference:
|
[7] Caginalp, G.: An analysis of a phase field model of a free boundary.Arch. Ration. Mech. Anal. 92 (1986), 205-245. Zbl 0608.35080, MR 0816623, 10.1007/BF00254827 |
| Reference:
|
[8] Cherfils, L., Miranville, A.: Some results on the asymptotic behavior of the Caginalp system with singular potentials.Adv. Math. Sci. Appl. 17 (2007), 107-129. Zbl 1145.35042, MR 2337372 |
| Reference:
|
[9] Chill, R., Fašangová, E., Prüss, J.: Convergence to steady states of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions.Math. Nachr. 279 (2006), 1448-1462. Zbl 1107.35058, MR 2269249, 10.1002/mana.200410431 |
| Reference:
|
[10] Fischer, H. P., Maass, P., Dieterich, W.: Novel surface modes in spinodal decomposition.Phys. Rev. Letters 79 (1997), 893-896. 10.1103/PhysRevLett.79.893 |
| Reference:
|
[11] Fischer, H. P., Maass, P., Dieterich, W.: Diverging time and length scales of spinodal decomposition modes in thin flows.Europhys. Letters 62 (1998), 49-54. 10.1209/epl/i1998-00550-y |
| Reference:
|
[12] Gal, C. G.: A Cahn-Hilliard model in bounded domains with permeable walls.Math. Methods Appl. Sci. 29 (2006), 2009-2036. Zbl 1113.35031, MR 2268279, 10.1002/mma.757 |
| Reference:
|
[13] Gal, C. G., Grasselli, M.: The non-isothermal Allen-Cahn equation with dynamic boundary conditions.Discrete Contin. Dyn. Syst. 22 (2008), 1009-1040. Zbl 1160.35353, MR 2434980, 10.3934/dcds.2008.22.1009 |
| Reference:
|
[14] Gatti, S., Miranville, A.: Asymptotic behavior of a phase-field system with dynamic boundary conditions.Differential Equations: Inverse and Direct Problems (Proceedings of the workshop "Evolution Equations: Inverse and Direct Problems", Cortona, June 21-25, 2004). A series of Lecture Notes in Pure and Applied Mathematics, Vol. 251 A. Favini and A. Lorenzi CRC Press Boca Raton (2006), 149-170. Zbl 1123.35310, MR 2275977 |
| Reference:
|
[15] Giorgi, C., Grasselli, M., Pata, V.: Uniform attractors for a phase-field model with memory and quadratic nonlinearity.Indiana Univ. Math. J. 48 (1999), 1395-1445. Zbl 0940.35037, MR 1757078, 10.1512/iumj.1999.48.1793 |
| Reference:
|
[16] Grasselli, M., Miranville, A., Pata, V., Zelik, S.: Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials.Math. Nachr. 280 (2007), 1475-1509. Zbl 1133.35017, MR 2354975, 10.1002/mana.200510560 |
| Reference:
|
[17] Grasselli, M., Petzeltová, H., Schimperna, G.: Long time behavior of solutions to the Caginalp system with singular potential.Z. Anal. Anwend. 25 (2006), 51-72. Zbl 1128.35021, MR 2216881, 10.4171/ZAA/1277 |
| Reference:
|
[18] Grasselli, M., Petzeltová, H., Schimperna, G.: Convergence to stationary solutions for a parabolic-hyperbolic phase-field system.Commun. Pure Appl. Anal. 5 (2006), 827-838. Zbl 1134.35017, MR 2246010, 10.3934/cpaa.2006.5.827 |
| Reference:
|
[19] Grasselli, M., Petzeltová, H., Schimperna, G.: A nonlocal phase-field system with inertial term.Q. Appl. Math. 65 (2007), 451-46. Zbl 1140.35352, MR 2354882, 10.1090/S0033-569X-07-01070-9 |
| Reference:
|
[20] Jendoubi, M. A.: A simple unified approach to some convergence theorems of L. Simon.J. Funct. Anal. 153 (1998), 187-202. Zbl 0895.35012, MR 1609269, 10.1006/jfan.1997.3174 |
| Reference:
|
[21] Kenzler, R., Eurich, F., Maass, P., Rinn, B., Schropp, J., Bohl, E., Dieterich, W.: Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions.Comput. Phys. Comm. 133 (2001), 139-157. Zbl 0985.65114, MR 1809807, 10.1016/S0010-4655(00)00159-4 |
| Reference:
|
[22] Łojasiewicz, S.: Ensembles semi-analytiques.IHES Bures-sur-Yvette (1965), French. |
| Reference:
|
[23] Miranville, A., Rougirel, A.: Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations.Z. Angew. Math. Phys. 57 (2006), 244-268. Zbl 1094.35102, MR 2214071, 10.1007/s00033-005-0017-6 |
| Reference:
|
[24] Miranville, A., Zelik, S.: Robust exponential attractors for singularly perturbed phase-field type equations.Electron. J. Differ. Equ. (2002), 1-28. Zbl 1004.35024, MR 1911930 |
| Reference:
|
[25] Miranville, A., Zelik, S.: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions.Math. Methods Appl. Sci. 28 (2005), 709-735. Zbl 1068.35020, MR 2125817, 10.1002/mma.590 |
| Reference:
|
[26] Prüss, J., Racke, R., Zheng, S.: Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions.Ann. Mat. Pura Appl. 185 (2006), 627-648. Zbl 1232.35081, MR 2230586, 10.1007/s10231-005-0175-3 |
| Reference:
|
[27] Prüss, J., Wilke, M.: Maximal $L_p$-regularity and long-time behaviour of the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions.Operator Theory: Advances and Applications, Vol. 168 Birkhäuser Basel (2006), 209-236. Zbl 1109.35060, MR 2240062 |
| Reference:
|
[28] Racke, R., Zheng, S.: The Cahn-Hilliard equation with dynamic boundary conditions.Adv. Diff. Equ. 8 (2003), 83-110. Zbl 1035.35050, MR 1946559 |
| Reference:
|
[29] Rybka, P., Hoffmann, K.-H.: Convergence of solutions to Cahn-Hilliard equation.Commun. Partial Differ. Equations 24 (1999), 1055-1077. Zbl 0936.35032, MR 1680877, 10.1080/03605309908821458 |
| Reference:
|
[30] Simon, L.: Asymptotics for a class of non-linear evolution equations, with applications to gemetric problems.Ann. Math. 118 (1983), 525-571. MR 0727703, 10.2307/2006981 |
| Reference:
|
[31] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition.Springer New York (1997). MR 1441312 |
| Reference:
|
[32] Wu, H., Zheng, S.: Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions.J. Differ. Equations 204 (2004), 511-531. Zbl 1068.35018, MR 2085545, 10.1016/j.jde.2004.05.004 |
| Reference:
|
[33] Zhang, Z.: Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions.Commun. Pure Appl. Anal. 4 (2005), 683-693. Zbl 1082.35033, MR 2167193, 10.3934/cpaa.2005.4.683 |
| Reference:
|
[34] Zhang, Z.: Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions.Commun. Pure Appl. Anal. 4 (2005), 683-693. Zbl 1082.35033, MR 2167193, 10.3934/cpaa.2005.4.683 |
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