×

Weak solutions for a degenerate phase-field model via Galerkin approximation. (English) Zbl 07869336

MSC:

35D30 Weak solutions to PDEs
35K65 Degenerate parabolic equations
74N20 Dynamics of phase boundaries in solids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI

References:

[1] J. W.Cahn and J. E.Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys.28 (1958), 258-267. · Zbl 1431.35066
[2] S. M.Allen and J. W.Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall.27 (1979), no. 6, 1085-1095.
[3] J. W.Cahn, On spinodal decomposition, Acta Metall.9 (1961), 795-801.
[4] F.Boyer and S.Minjeaud, Hierarchy of consistent n‐component Cahn‐Hilliard system, Math. Models Methods Appl. Sci.24 (2014), 2885-2928. · Zbl 1308.35004
[5] J.Verdasca, P.Borckmans, and G.Dewel, Chemically frozen phase separation in an adsorbed layer, Phys. Rev. E52 (1995), 4616-4619.
[6] P.Colli, G.Gilardi, and D.Hilhorst, On a Cahn‐Hilliard type phase field model related to tumor growth, Discrete Contin. Dyn. Syst.35 (2015), 2423-2442. · Zbl 1342.35407
[7] P.Colli, G.Gilardi, E.Rocca, and J.Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity30 (2017), 2518-2546. · Zbl 1378.35175
[8] R.Kobayashi, Modelling and numerical simulations of dendritic crystal growth, Phys. D63 (1993), 410-423. · Zbl 0797.35175
[9] H. D.Alber and P. C.Zhu, Evolution of phase boundaries by configurational forces, Arch. Rational Mech. Anal.185 (2007), 235-286. · Zbl 1117.74045
[10] H. D.Alber and P. C.Zhu, Solutions to a model for interface motion by interface diffusion, Proc. Roy. Soc. Edinburgh138 (2008), 923-955. · Zbl 1168.35024
[11] H. D.Alber and P. C.Zhu, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. Appl. Math.66 (2005), no. 2, 680-699. · Zbl 1096.35068
[12] L. X.Zhao and H.Cheng, Global weak solutions to the 1D phase‐field model with inhomogeneous elasticity, Appl. Math. Model.104 (2022), 567-586. · Zbl 1505.74172
[13] X.Han and X. Z.Bian, Viscosity solutions to a new phase‐field model with Neumann boundary condition for solid‐solid phase transitions, J. Math. Anal. Appl.486 (2020), no. 2, 123900. · Zbl 1444.35091
[14] W. C.Sheng and P. C.Zhu, Viscosity solutions to a model for solid‐solid phase transitions driven by material forces, Nonlinear Anal. Real World Appl.39 (2018), 14-32. · Zbl 1462.74127
[15] P. C.Zhu, Solvability via viscosity solutions for a model of phase transitions driven by configurational forces, J. Differential Equations251 (2011), no. 10, 2833-2852. · Zbl 1262.35196
[16] P. C.Zhu, Regularity of solutions to a model for solid‐solid phase transitions driven by configurational forces, J. Math. Anal. Appl.389 (2012), no. 2, 1159-1172. · Zbl 1234.35053
[17] S.Kawashima and P. C.Zhu, Traveling waves for models of phase transitions of solids driven by configurational forces, Discrete Contin. Dyn. Syst.15 (2011), no. 1, 309-323. · Zbl 1219.35036
[18] H. D.Alber, Evolving microstructure and homogenization, Contin. Mech. Thermodyn.12 (2000), 235-286. · Zbl 0983.74045
[19] X. Z.Bian and L. P.Luan, Global solutions to a model with Dirichlet boundary conditions for interface motion by interface diffusion, J. Math. Phys.61 (2020), 41503. · Zbl 1446.82059
[20] J. L.Lions, Quelques Methodes De Resolution Des Problemes Aux Limites Non Lineaires, Dunod Gauthier‐Villars, Paris, 1969. · Zbl 0189.40603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.