# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022139
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## Weak solutions to an initial-boundary value problem for a continuum equation of motion of grain boundaries

 1 Department of Mathematics, Shanghai University, Shanghai 200444, China 2 Department of Mathematics, Hong Kong University of Science and Technology, Clear water bay, Kowloon, Hong Kong 3 HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute, Futian, Shenzhen, China

*Corresponding author: Yang Xiang

Received  April 2022 Revised  June 2022 Early access August 2022

Fund Project: The work of P. C. Zhu was supported in part by Science and Technology Commission of Shanghai Municipality (Grant No. 20JC1413600). The work of Y. Xiang was supported by the Hong Kong Research Grants Council General Research Fund 16302818 and Collaborative Research Fund C1005-19G, and the Project of Hetao Shenzhen-HKUST Innovation Cooperation Zone HZQB-KCZYB-2020083

We investigate an initial-(periodic-)boundary value problem for a continuum equation, which is a model for motion of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and integrated the effects of a diverse range of thermodynamic driving forces. We first prove the global-in-time existence and uniqueness of weak solution to this initial-boundary value problem in the case with positive equilibrium disconnection density parameter $B$, and then investigate the asymptotic behavior of the solutions as $B$ goes to zero. The main difficulties in the proof of main theorems are due to the degeneracy of $B=0$, a non-local term with singularity, and a non-smooth coefficient of the highest derivative associated with the gradient of the unknown. The key ingredients in the proof are the energy method, an estimate for a singular integral of the Hilbert type, and a compactness lemma.

Citation: Peicheng Zhu, Lei Yu, Yang Xiang. Weak solutions to an initial-boundary value problem for a continuum equation of motion of grain boundaries. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022139
##### References:
 [1] A. Acharya, K. Matthies and J. Zimmer, Traveling wave solutions for a quasilinear model of field dislocation mechanics, J. Mech. Phys. Solids., 58 (2010), 2043-2053.  doi: 10.1016/j.jmps.2010.09.008. [2] H.-D. Alber and P. Zhu, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. Appl. Math., 66 (2006), 680-699.  doi: 10.1137/050629951. [3] H.-D. Alber and P. Zhu, Evolution of phase boundaries by configurational forces, Arch. Rati. Mech. Anal., 185 (2007), 235-286.  doi: 10.1007/s00205-007-0054-8. [4] H.-D. Alber and P. Zhu, Solutions to a model for interface motion by interface diffusion, Proc. Roy. Soc. Edin., A138 (2008), 923-955.  doi: 10.1017/S0308210507000170. [5] H.-D. Alber and P. Zhu, Interface motion by interface diffusion driven by bulk energy: Justification of a diffusive interface model, Conti. Mech. Thermodyn., 23 (2011), 139-176.  doi: 10.1007/s00161-010-0162-9. [6] H.-D. Alber and P. Zhu, Solutions to a model with neumann boundary conditions for phase transitions driven by configurational forces, Nonlinear Anal. RWA., 12 (2011), 1797-1809.  doi: 10.1016/j.nonrwa.2010.11.012. [7] M. F. Ashby, Boundary defects, and atomistic aspects of boundary sliding and diffusional creep, Surf. Sci., 31 (1972), 498-542. [8] J. W. Cahn and S. M. Allen, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095. [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [10] J. W. Cahn and J. E. Taylor, A unified approach to motion of grain boundaries, relative tangential translation along grain boundaries, and grain rotation, Acta Mater., 52 (2004), 4887-4898. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der Mathematischen Wissenschaften Vol. 224, Springer-Verlag Berlin Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0. [12] F. Hildebrand and C. Miehe, A regularized sharp-interface model for phase transformation accounting for prescribed sharp-interface kinetics, Proc. Appl. Math. Mech., 10 (2010), 673-676. [13] J. P. Hirth and J. Lothe, Theory of Dislocations, , 2nd edition, John Wiley, New York, 1982. [14] J. P. Hirth and R. C. Pond, Steps, dislocations and disconnections as interface defects relating to structure and phase transformations, Acta Mater., 44 (1996), 4749-4763. [15] J. P. Hirth, R. C. Pond and J. Lothe, Spacing defects and disconnections in grain boundaries, Acta Mater., 55 (2007), 5428-5437. [16] K. G. F. Janssens, D. Olmsted, E. A. Holm, S. M. Foiles, S. J. Plimpton and P. M. Derlet, Computing the mobility of grain boundaries, Nat. Mater., 5 (2006), 124-127. [17] A. H. King and D. A. Smith, The effects on grain-boundary processes of the steps in the boundary plane associated with the cores of grain-boundary dislocations, Acta Cryst., A36 (1980), 335-343. [18] O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'Tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. [19] C. H. Li, E. H. Edwards, J. Washburn and E. R. Parker, Stress-induced movement of crystal boundaries, Acta Metall., 1 (1953), 223-229. [20] J. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod Gauthier-Villars, Paris, 1969. [21] K. L. Merkle, L. J. Thompson and F. Phillipp, In-Situ HREM Studies of Grain Boundary Migration, Interface Sci., 12 (2004), 277-292. [22] A. Rajabzadeh, M. Legros, N. Combe, F. Mompiou and D. A. Molodov, Evidence of grain boundary dislocation step motion associated to shear-coupled grain boundary migration, Phil. Mag., 93 (2013), 1299-1316. [23] A. Rajabzadeh, F. Mompiou, M. Legros and N. Combe, Elementary mechanisms of shear-coupled grain boundary migration, Phys. Rev. Lett., 110 (2013), 265507. [24] T. Roubiček, A generalization of the Lions-Temam compact imbedding theorem, Časopis pro Pěst. Mat., 115 (1990), 338–342. [25] J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. [27] A. P. Sutton and R. W. Balluffi, Interfaces in Crystalline Materials, Oxford University Press, New York, 1995. [28] S. Thomas, K. Chen, J. Han, P. Purohit and D. J. Srolovitz, Reconciling grain growth and shear-coupled grain boundary migration, Nature Commun., 8 (2017), 1764. [29] C. Wei, S. Thomas, J. Han, D. J. Srolovitz and Y. Xiang, A continuum multi-disconnection-mode model for grain boundary migration, J. Mech. Phys. Solids, 133 (2019), 103731.  doi: 10.1016/j.jmps.2019.103731. [30] C. Wei, L. Zhang, J. Han, D. J. Srolovitz and Y. Xiang, Grain boundary triple junction dynamics: A continuum disconnection model, SIAM J. Appl. Math., 80 (2020), 1101-1122.  doi: 10.1137/19M1277722. [31] M. Winning, G. Gottstein and L. Shvindlerman, Stress induced grain boundary motion, Acta Mater., 49 (2001), 211-219. [32] Y. Xiang, Modeling dislocations at different scales, Commun. Comput. Phys., 1 (2006), 383-424. [33] L. Zhang, J. Han, D. J. Srolovitz and Y. Xiang, Equation of motion for grain boundaries in polycrystals, npj Comput. Mater., 7 (2021), 64. [34] L. Zhang, J. Han, Y. Xiang and D. J. Srolovitz, Equation of motion for a grain boundary, Phys. Rev. Lett., 119 (2017), 246101. [35] P. Zhu, Solvability via viscosity solutions for a model of phase transitions driven by configurational forces, J. Diff. Eqs., 251 (2011), 2833-2852.  doi: 10.1016/j.jde.2011.05.035. [36] P. Zhu, Solid-Solid Phase Transitions Driven by Configurational Forces: A phase-Field Model and Its Validity, LAMBERT Academic Publishing (LAP), July, 2011. [37] P. Zhu, Regularity of solutions to a model for solid phase transitions driven by configurational forces, J. Math. Anal. Appl., 389 (2012), 1159-1172.  doi: 10.1016/j.jmaa.2011.12.052.

show all references

##### References:
 [1] A. Acharya, K. Matthies and J. Zimmer, Traveling wave solutions for a quasilinear model of field dislocation mechanics, J. Mech. Phys. Solids., 58 (2010), 2043-2053.  doi: 10.1016/j.jmps.2010.09.008. [2] H.-D. Alber and P. Zhu, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. Appl. Math., 66 (2006), 680-699.  doi: 10.1137/050629951. [3] H.-D. Alber and P. Zhu, Evolution of phase boundaries by configurational forces, Arch. Rati. Mech. Anal., 185 (2007), 235-286.  doi: 10.1007/s00205-007-0054-8. [4] H.-D. Alber and P. Zhu, Solutions to a model for interface motion by interface diffusion, Proc. Roy. Soc. Edin., A138 (2008), 923-955.  doi: 10.1017/S0308210507000170. [5] H.-D. Alber and P. Zhu, Interface motion by interface diffusion driven by bulk energy: Justification of a diffusive interface model, Conti. Mech. Thermodyn., 23 (2011), 139-176.  doi: 10.1007/s00161-010-0162-9. [6] H.-D. Alber and P. Zhu, Solutions to a model with neumann boundary conditions for phase transitions driven by configurational forces, Nonlinear Anal. RWA., 12 (2011), 1797-1809.  doi: 10.1016/j.nonrwa.2010.11.012. [7] M. F. Ashby, Boundary defects, and atomistic aspects of boundary sliding and diffusional creep, Surf. Sci., 31 (1972), 498-542. [8] J. W. Cahn and S. M. Allen, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095. [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [10] J. W. Cahn and J. E. Taylor, A unified approach to motion of grain boundaries, relative tangential translation along grain boundaries, and grain rotation, Acta Mater., 52 (2004), 4887-4898. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der Mathematischen Wissenschaften Vol. 224, Springer-Verlag Berlin Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0. [12] F. Hildebrand and C. Miehe, A regularized sharp-interface model for phase transformation accounting for prescribed sharp-interface kinetics, Proc. Appl. Math. Mech., 10 (2010), 673-676. [13] J. P. Hirth and J. Lothe, Theory of Dislocations, , 2nd edition, John Wiley, New York, 1982. [14] J. P. Hirth and R. C. Pond, Steps, dislocations and disconnections as interface defects relating to structure and phase transformations, Acta Mater., 44 (1996), 4749-4763. [15] J. P. Hirth, R. C. Pond and J. Lothe, Spacing defects and disconnections in grain boundaries, Acta Mater., 55 (2007), 5428-5437. [16] K. G. F. Janssens, D. Olmsted, E. A. Holm, S. M. Foiles, S. J. Plimpton and P. M. Derlet, Computing the mobility of grain boundaries, Nat. Mater., 5 (2006), 124-127. [17] A. H. King and D. A. Smith, The effects on grain-boundary processes of the steps in the boundary plane associated with the cores of grain-boundary dislocations, Acta Cryst., A36 (1980), 335-343. [18] O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'Tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. [19] C. H. Li, E. H. Edwards, J. Washburn and E. R. Parker, Stress-induced movement of crystal boundaries, Acta Metall., 1 (1953), 223-229. [20] J. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod Gauthier-Villars, Paris, 1969. [21] K. L. Merkle, L. J. Thompson and F. Phillipp, In-Situ HREM Studies of Grain Boundary Migration, Interface Sci., 12 (2004), 277-292. [22] A. Rajabzadeh, M. Legros, N. Combe, F. Mompiou and D. A. Molodov, Evidence of grain boundary dislocation step motion associated to shear-coupled grain boundary migration, Phil. Mag., 93 (2013), 1299-1316. [23] A. Rajabzadeh, F. Mompiou, M. Legros and N. Combe, Elementary mechanisms of shear-coupled grain boundary migration, Phys. Rev. Lett., 110 (2013), 265507. [24] T. Roubiček, A generalization of the Lions-Temam compact imbedding theorem, Časopis pro Pěst. Mat., 115 (1990), 338–342. [25] J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. [27] A. P. Sutton and R. W. Balluffi, Interfaces in Crystalline Materials, Oxford University Press, New York, 1995. [28] S. Thomas, K. Chen, J. Han, P. Purohit and D. J. Srolovitz, Reconciling grain growth and shear-coupled grain boundary migration, Nature Commun., 8 (2017), 1764. [29] C. Wei, S. Thomas, J. Han, D. J. Srolovitz and Y. Xiang, A continuum multi-disconnection-mode model for grain boundary migration, J. Mech. Phys. Solids, 133 (2019), 103731.  doi: 10.1016/j.jmps.2019.103731. [30] C. Wei, L. Zhang, J. Han, D. J. Srolovitz and Y. Xiang, Grain boundary triple junction dynamics: A continuum disconnection model, SIAM J. Appl. Math., 80 (2020), 1101-1122.  doi: 10.1137/19M1277722. [31] M. Winning, G. Gottstein and L. Shvindlerman, Stress induced grain boundary motion, Acta Mater., 49 (2001), 211-219. [32] Y. Xiang, Modeling dislocations at different scales, Commun. Comput. Phys., 1 (2006), 383-424. [33] L. Zhang, J. Han, D. J. Srolovitz and Y. Xiang, Equation of motion for grain boundaries in polycrystals, npj Comput. Mater., 7 (2021), 64. [34] L. Zhang, J. Han, Y. Xiang and D. J. Srolovitz, Equation of motion for a grain boundary, Phys. Rev. Lett., 119 (2017), 246101. [35] P. Zhu, Solvability via viscosity solutions for a model of phase transitions driven by configurational forces, J. Diff. Eqs., 251 (2011), 2833-2852.  doi: 10.1016/j.jde.2011.05.035. [36] P. Zhu, Solid-Solid Phase Transitions Driven by Configurational Forces: A phase-Field Model and Its Validity, LAMBERT Academic Publishing (LAP), July, 2011. [37] P. Zhu, Regularity of solutions to a model for solid phase transitions driven by configurational forces, J. Math. Anal. Appl., 389 (2012), 1159-1172.  doi: 10.1016/j.jmaa.2011.12.052.
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