We employ the Galerkin method to prove the global existence of weak solutions to a phase-field model which is suitable to describe a sort of interface motion driven by configurational forces. The higher-order derivative of unknown S exists in the sense of local weak derivatives since it may be not summable over the original open domain. The existence proof is valid in the one-dimensional case.
Zixian ZHU, Boling GUO, Shaomei FANG
. Global weak solutions to a phase-field model for motion of grain boundaries[J]. Applied Mathematics and Mechanics, 2022
, 43(11)
: 1777
-1792
.
DOI: 10.1007/s10483-022-2915-9
[1] HERRING, C. Surface tension as a motivation for sintering. The Physics of Powder Metallurgy, McGrawHill Book Company, New York, 143-179(1951)
[2] CAHN, J. W. and HILLIARD, J. E. Free energy of a nonuniform system, I, interfacial free energy. The Journal of Chemical Physics, 28(2), 258-267(1958)
[3] ALBER, H. D. and ZHU, P. C. Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces. SIAM Journal on Applied Mathematics, 66(2), 680-699(2005)
[4] ALBER, H. D. and ZHU, P. C. Evolution of phase boundaries by configurational forces. Archive for Rational Mechanics and Analysis, 185, 235-286(2007)
[5] ALBER, H. D. and ZHU, P. C. Solutions to a model for interface motion by interface diffusion. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 138(5), 923-955(2008)
[6] ZHU, P. C. Solid-Solid Phase Transitions Driven by Configurational Forces:A Phase-Field Model and its Validity, Lambert Academy Publishing (LAP), Germany (2011)
[7] BIAN, X. Z. and LUAN, L. P. Global solutions to a model with Dirichlet boundary conditions for interface motion by interface diffusion. Journal of Mathematical Physics, 61, 041503(2020)
[8] WU, F., BIAN, X. Z., and ZHAO, L. X. Solutions to a phase-field model for martensitic phase transformations driven by configurational forces. Studies in Applied Mathematics, 146(3), 730-752(2021)
[9] ZHAO, L. X., MAI, J. W., and CHENG, F. Analysis for a new phase-field model with periodic boundary conditions. Mathematical Methods in the Applied Sciences, 44(5), 3786-3803(2020)
[10] ZHAO, L. X., MAI, J. W., and CHENG, F. Weak solutions and simulations to a new phase-field model with periodic boundary. Applicable Analysis, 101(1), 366-381(2020)
[11] HAN, X. and BIAN, X. Z. Viscosity solutions to a new phase-field model with Neumann boundary condition for solid-solid phase transitions. Journal of Mathematical Analysis and Applications, 486(2), 123900(2020)
[12] SHENG, W. C. and ZHU, P. C. Viscosity solutions to a model for solid-solid phase transitions driven by material forces. Nonlinear Analysis:Real World Applications, 39, 14-32(2018)
[13] ZHENG, J. Z. Viscosity solutions to a Cauchy problem of a phase-field model for solid-solid phase transitions. Studies in Applied Mathematics, 145(3), 483-499(2020).
[14] ZHU, P. C. Solvability via viscosity solutions for a model of phase transitions driven by configurational forces. Journal of Differential Equations, 251(10), 2833-2852(2011)
[15] ZHU, P. C. Regularity of solutions to a model for solid-solid phase transitions driven by configurational forces. Journal of Mathematical Analysis and Applications, 389(2), 1159-1172(2012)
[16] EVANS, L. C. Partial Differential Equations, 2nd ed., American Mathematical Society, the United States of America, 355-356(2010)
[17] NIRENBERG, L. On elliptic partial differential equations. Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, 13, 115-162(1959)
[18] LIONS, J. L. Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod Gauthier-Villars, Paris (1969)
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