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Solutions to a model with Neumann boundary conditions for phase transitions driven by configurational forces. (English) Zbl 1215.35012
Summary: We study an initial boundary value problem of a model describing the evolution in time of diffusive phase interfaces in solid materials, in which martensitic phase transformations driven by configurational forces take place. The model was proposed earlier by the authors and consists of the partial differential equations of linear elasticity coupled to a nonlinear, degenerate parabolic equation of second order for an order parameter. In a previous paper global existence of weak solutions in one space dimension was proved under Dirichlet boundary conditions for the order parameter. Here we show that global solutions also exist for Neumann boundary conditions. Again, the method of proof is only valid in one space dimension.

MSC:
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35K65 Degenerate parabolic equations
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[1] Alber, H.-D.; Zhu, Peicheng, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. appl. math., 66, 2, 680-699, (2006) · Zbl 1096.35068
[2] Alber, H.-D., Evolving microstructure and homogenization, Contin. mech. thermodyn., 12, 235-287, (2000) · Zbl 0983.74045
[3] Alber, H.-D.; Zhu, Peicheng, Evolution of phase boundaries by configurational forces, Arch. rational mech. anal., 185, 235-286, (2007) · Zbl 1117.74045
[4] Abeyaratne, R.; Knowles, J.K., On the driving traction acting on a surface of strain discontinuity in a continuum, J. mech. phys. solids, 38, 3, 345-360, (1990) · Zbl 0713.73030
[5] Ladyzenskaya, O.; Solonnikov, V.; Uralceva, N., Linear and quasilinear equations of parabolic type, ()
[6] H.-D. Alber, Peicheng Zhu, Precise asymptotic expansions for solutions of phase field models at the passage to the sharp interface limit, 2010, Manuscript.
[7] Bonetti, E.; Colli, P.; Dreyer, W.; Gilardi, G.; Schimpanera, G.; Sprekels, J., On a model for phase separation in binary alloys driven by mechanical effects, Physica D, 165, 48-65, (2002) · Zbl 1008.74066
[8] Buratti, G.; Huo, Y.; Müller, I., Eshelby tensor as a tensor of free enthalpy, J. elasticity, 72, 31-42, (2003) · Zbl 1059.74046
[9] Carrive, M.; Miranville, A.; Pierus, A., The cahn – hilliard equation for deformable elastic continua, Adv. math. sci. appl., 10, 2, 539-569, (2000) · Zbl 0987.35156
[10] Garcke, H., On cahn – hilliard systems with elasticity, Proc. R. soc. Edinburgh A, 133, 2, 307-331, (2003) · Zbl 1130.74037
[11] Hornbogen, E.; Warlimont, H., Metallkunde, (2001), Springer-Verlag 4. Auflage
[12] James, R., Configurational forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever, Contin. mech. thermodyn., 14, 55-86, (2002) · Zbl 1100.74549
[13] Alber, H.-D.; Zhu, Peicheng, Solutions to a model for interface motion by interface diffusion, Proc. roy. soc. Edinburgh, 138A, 923-955, (2008) · Zbl 1168.35024
[14] H.-D. Alber, Peicheng Zhu, Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model, Contin. Mech. Thermodyn., online first 2010, http://dx.doi.org/10.1007/s00161-010-0162-9 (2010).
[15] Fried, E.; Gurtin, M., Dynamic solid-solid transitions with phase characterized by an order parameter, Physica D, 72, 287-308, (1994) · Zbl 0812.35164
[16] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. mat. pura appl., 146, 64-96, (1987) · Zbl 0629.46031
[17] Roubícěk, T., A generalization of the Lions-temam compact embedding theorem, Cas. pest. mat., 115, 338-342, (1990) · Zbl 0755.46013
[18] Lions, J., Quelques methodes de resolution des problemes aux limites non lineaires, (1969), Dunod Gauthier-Villars Paris · Zbl 0189.40603
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