Previous |  Up |  Next

Article

Keywords:
nonisothermal Ginzburg-Landau (Allen-Cahn) system; microforce balance; existence and uniqueness results; renormalized solutions; Moser iterations
Summary:
The article deals with a nonlinear generalized Ginzburg-Landau (Allen-Cahn) system of PDEs accounting for nonisothermal phase transition phenomena which was recently derived by A. Miranville and G. Schimperna: Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst., Ser. B, {\it 5} (2005), 753--768. The existence of solutions to a related Neumann-Robin problem is established in an $N \le 3$-dimensional space setting. A fixed point procedure guarantees the existence of solutions locally in time. Next, Sobolev embeddings, interpolation inequalities, Moser iterations estimates and results on renormalized solutions for a parabolic equation with $L^1$ data are used to handle a suitable a priori estimate which allows to extend our local solutions to the whole time interval. The uniqueness result is justified by proper contracting estimates.
References:
[1] Adams, R. A.: Sobolev Spaces. Academic Press New York (1975). MR 0450957 | Zbl 0314.46030
[2] Adams, R. A., Fournier, J.: Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61 (1977), 713-734. DOI 10.1016/0022-247X(77)90173-1 | MR 0463902 | Zbl 0385.46024
[3] Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, II. Commun. Pure Appl. Math. 12 (1959), 623-727 17 (1964), 35-92. DOI 10.1002/cpa.3160120405 | MR 0162050
[4] Alt, H. W., Pawlow, I.: A mathematical model of dynamics of non-isothermal phase separation. Physica D 59 (1992), 389-416. DOI 10.1016/0167-2789(92)90078-2 | MR 1192751 | Zbl 0763.58031
[5] Attouch, H.: Variational Convergence for Functions and Operators. Pitman London (1984). MR 0773850 | Zbl 0561.49012
[6] Baiocchi, C.: Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert. Ann. Mat. Pura Appl. 76 (1967), 233-304 Italian. DOI 10.1007/BF02412236 | MR 0223697 | Zbl 0153.17202
[7] Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff Leyden (1976). MR 0390843 | Zbl 0328.47035
[8] Blanchard, D., Francfort, G. A.: A few results on a class of degenerate parabolic equations. Ann. Sc. Norm. Sup. Pisa 18 (1991), 213-249. MR 1129302 | Zbl 0778.35046
[9] Blanchard, D., Guibé, O.: Existence of a solution for a nonlinear system in thermoviscoelasticity. Adv. Differ. Equ. 5 (2000), 1221-1252. MR 1785674
[10] Blanchard, D., Redwane, H.: Renormalized solutions for a class of nonlinear evolution problems. J. Math. Pures Appl. 77 (1998), 117-151. DOI 10.1016/S0021-7824(98)80067-6 | MR 1614645 | Zbl 0907.35070
[11] Bonfanti, G., Frémond, M., Luterotti, F.: Global solution to a nonlinear system for inversible phase changes. Adv. Math. Sci. Appl. 10 (2000), 1-24. MR 1769184
[12] Bonfanti, G., Frémond, M., Luterotti, F.: Local solutions to the full model of phase transitions with dissipation. Adv. Math. Sci. Appl. 11 (2001), 791-810. MR 1907467
[13] Bonfanti, G., Frémond, M., Luterotti, F.: Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements. Nonlinear Anal., Real World Appl. 5 (2004), 123-140. MR 2004090
[14] Bonfanti, G., Luterotti, F.: Global solution to a phase transition model with microscopic movements and accelerations in one space dimension. Commun. Pure Appl. Anal. 5 (2006), 763-777. DOI 10.3934/cpaa.2006.5.763 | MR 2246006 | Zbl 1137.80010
[15] Brézis, H.: Analyse fonctionnelle. Théorie et applications. Masson Paris (1983), French. MR 0697382
[16] Colli, P.: On some doubly nonlinear evolution equations in Banach spaces. Japan J. Ind. Appl. Math. 9 (1992), 181-203. DOI 10.1007/BF03167565 | MR 1170721 | Zbl 0757.34051
[17] Colli, P., Frémond, M., Klein, O.: Global existence of a solution to phase field model for supercooling. Nonlinear Anal., Real World Appl. 2 (2001), 523-539. MR 1858904
[18] Colli, P., Gilardi, G., Grasselli, M.: Well-posedness of the weak formulation for the phase-field model with memory. Adv. Differ. Equ. 2 (1997), 487-508. MR 1441853 | Zbl 1023.45501
[19] Colli, P., Gilardi, G., Grasselli, M., Schimperna, G.: Global existence for the conserved phase field model with memory and quadratic nonlinearity. Port. Math. (N.S.) 58 (2001), 159-170. MR 1836260 | Zbl 0985.35094
[20] Colli, P., Laurençot, Ph.: Existence and stabilization of solutions to the phase-field model with memory. J. Integral Equations Appl. 10 (1998), 169-194. DOI 10.1216/jiea/1181074220 | MR 1646829
[21] Colli, P., Luterotti, F., Schimperna, G., Stefanelli, U.: Global existence for a class of generalized systems for irreversible phase changes. NoDEA, Nonlinear Differ. Equ. Appl. 9 (2002), 255-276. DOI 10.1007/s00030-002-8127-8 | MR 1917373 | Zbl 1004.35061
[22] Damlamian, A.: Some results on the multi-phase Stefan problem. Commun. Partial Differ. Equations 2 (1977), 1017-1044. DOI 10.1080/03605307708820053 | MR 0487015 | Zbl 0399.35054
[23] Damlamian, A., Kenmochi, N.: Evolution equations generated by subdifferentials in the dual space of $H^1(\Omega)$. Discrete Contin. Dyn. Syst. 5 (1999), 269-278. DOI 10.3934/dcds.1999.5.269 | MR 1665795
[24] DiPerna, R. J., Lions, J.-L.: On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130 (1989), 321-366. DOI 10.2307/1971423 | MR 1014927 | Zbl 0698.45010
[25] DiPerna, R. J., Lions, J.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511-547. DOI 10.1007/BF01393835 | MR 1022305 | Zbl 0696.34049
[26] Frémond, M.: Non-Smooth Thermomechanics. Springer Berlin (2002). MR 1885252
[27] Gurtin, M.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on microforce balance. Physica D 92 (1996), 178-192. DOI 10.1016/0167-2789(95)00173-5 | MR 1387065
[28] Ladyzhenskaya, O. A., Solonnikov, V. A., Ural'tseva, N. N.: Linear and Quasi-linear Equations of Parabolic Type. Translation of Mathematical Monographs, 23. AMS Providence (1968).
[29] Laurençot, Ph., Schimperna, G., Stefanelli, U.: Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions. J. Math. Anal. Appl. 271 (2002), 426-442. DOI 10.1016/S0022-247X(02)00127-0 | MR 1923644
[30] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod/Gauthier-Villars Paris (1969), French. MR 0259693 | Zbl 0189.40603
[31] Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications. Dunod Paris (1968), French. Zbl 0165.10801
[32] Luterotti, F., Stefanelli, U.: Existence result for the one-dimensional full model of phase transitions. Z. Anal. Anwend. 21 (2002), 335-350. DOI 10.4171/ZAA/1081 | MR 1915265 | Zbl 1003.80003
[33] Luterotti, F., Schimperna, G., Stefanelli, U.: Existence result for a nonlinear model related to irreversible phase changes. Math. Models Methods Appl. Sci. 11 (2001), 808-825. DOI 10.1142/S0218202501001112 | MR 1842227 | Zbl 1013.35045
[34] Luterotti, F., Schimperna, G., Stefanelli, U.: Global solution to a phase field model with irreversible and constrained phase evolution. Q. Appl. Math. 60 (2002), 301-316. MR 1900495 | Zbl 1032.35109
[35] Luterotti, F., Schimperna, G., Stefanelli, U.: Local solution to Frémond's full model for irreversible phase transitions. In: Mathematical Models and Methods for Smart Materials. Proc. Conf., Cortona, Italy, June 25-29, 2001 M. Fabrizio, B. Lazzari, A. Mauro World Scientific River Edge (2002), 323-328. MR 2039276 | Zbl 1049.35096
[36] Miranville, A., Schimperna, G.: Nonisothermal phase separation based on a microforce balance. Discrete Contin. Dyn. Syst., Ser. B 5 (2005), 753-768. DOI 10.3934/dcdsb.2005.5.753 | MR 2151731 | Zbl 1140.80388
[37] Miranville, A., Schimperna, G.: Global solution to a phase transition model based on a microforce balance. J. Evol. Equ. 5 (2005), 253-276. DOI 10.1007/s00028-005-0187-x | MR 2133444 | Zbl 1074.35050
[38] Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa, III. Ser. 123 (1959), 115-162. MR 0109940 | Zbl 0088.07601
[39] Rakotoson, J. E., Rakotoson, J. M.: Analyse fonctionnelle appliquée aux équations aux dérivées partielles. Presse Universitaires de France (1999), French. MR 1686529 | Zbl 0929.46027
[40] Simon, J.: Compact sets in the space $L^p(0 , T ; B)$. Ann. Mat. Pura Appl., IV. Ser. 146 (1978), 65-96. DOI 10.1007/BF01762360 | MR 0916688
[41] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer New York (1988). MR 0953967 | Zbl 0662.35001
Partner of
EuDML logo