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On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation
Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion
1. | Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 |
2. | Department of General Education, Salesian Polytechnic, 4-6-8 Oyamagaoka, Machida-city, Tokyo, 194-0215, Japan |
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'' Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000. |
[2] |
F. Andreu, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics, 223, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/978-3-0348-7928-6. |
[3] |
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,'' Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, RI, 2010. |
[4] |
F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Local and nonlocal weighted $ p $-Laplacian evolution equations with Neumann boundary conditions, Publ. Mat., 55 (2011), 27-66.
doi: 10.5565/PUBLMAT_55111_03. |
[5] |
G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4), 135 (1983), 293-318.
doi: 10.1007/BF01781073. |
[6] |
H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces,'' Applications to PDEs and Optimization, MPS-SIAM Series on Optimization, SIAM and MPS, 2001. |
[7] |
H. Brézis, "Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,'' North-Holland Mathematics Studies, 5, Notas de Matemática (50), North-Holland Publishing and American Elsevier Publishing, 1973. |
[8] |
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'' Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992. |
[9] |
M. -H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Jpn. J. Ind. Appl. Math., 27 (2010), 323-345.
doi: 10.1007/s13160-010-0020-y. |
[10] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984. |
[11] |
A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433-454.
doi: 10.1007/s10492-008-0035-8. |
[12] |
A. Ito, N. Kenmochi and N. Yamazaki, Global solvability of a model for grain boundary motion with constraint, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 127-146. |
[13] |
N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Available from: http://ci.nii.ac.jp/naid/110004715232. |
[14] |
R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220.
doi: 10.1023/A:1004570921372. |
[15] |
R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundary, Phys. D, 140 (2000), 141-150.
doi: 10.1016/S0167-2789(00)00023-3. |
[16] |
R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, in "Free Boundary Problems: Theory and Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 283-294. |
[17] |
J. S. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218.
doi: 10.1007/s00208-004-0624-0. |
[18] |
U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585.
doi: 10.1016/0001-8708(69)90009-7. |
[19] |
M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators: Cauchy problems, J. Differential Equations, 46 (1982), 268-299.
doi: 10.1016/0022-0396(82)90119-X. |
[20] |
K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions,'' Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ, (2006), 269-288.
doi: 10.1142/9789812774293_0014. |
[21] |
K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability for one-dimensional phase field system associated with grain boundary motion, Math. Ann., 356 (2013), 301-330.
doi: 10.1007/s00208-012-0849-2. |
[22] |
J. Simon, Compact set in the space $ L^p(0, T; B) $, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'' Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000. |
[2] |
F. Andreu, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics, 223, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/978-3-0348-7928-6. |
[3] |
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,'' Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, RI, 2010. |
[4] |
F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Local and nonlocal weighted $ p $-Laplacian evolution equations with Neumann boundary conditions, Publ. Mat., 55 (2011), 27-66.
doi: 10.5565/PUBLMAT_55111_03. |
[5] |
G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4), 135 (1983), 293-318.
doi: 10.1007/BF01781073. |
[6] |
H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces,'' Applications to PDEs and Optimization, MPS-SIAM Series on Optimization, SIAM and MPS, 2001. |
[7] |
H. Brézis, "Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,'' North-Holland Mathematics Studies, 5, Notas de Matemática (50), North-Holland Publishing and American Elsevier Publishing, 1973. |
[8] |
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'' Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992. |
[9] |
M. -H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Jpn. J. Ind. Appl. Math., 27 (2010), 323-345.
doi: 10.1007/s13160-010-0020-y. |
[10] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984. |
[11] |
A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433-454.
doi: 10.1007/s10492-008-0035-8. |
[12] |
A. Ito, N. Kenmochi and N. Yamazaki, Global solvability of a model for grain boundary motion with constraint, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 127-146. |
[13] |
N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Available from: http://ci.nii.ac.jp/naid/110004715232. |
[14] |
R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220.
doi: 10.1023/A:1004570921372. |
[15] |
R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundary, Phys. D, 140 (2000), 141-150.
doi: 10.1016/S0167-2789(00)00023-3. |
[16] |
R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, in "Free Boundary Problems: Theory and Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 283-294. |
[17] |
J. S. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218.
doi: 10.1007/s00208-004-0624-0. |
[18] |
U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585.
doi: 10.1016/0001-8708(69)90009-7. |
[19] |
M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators: Cauchy problems, J. Differential Equations, 46 (1982), 268-299.
doi: 10.1016/0022-0396(82)90119-X. |
[20] |
K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions,'' Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ, (2006), 269-288.
doi: 10.1142/9789812774293_0014. |
[21] |
K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability for one-dimensional phase field system associated with grain boundary motion, Math. Ann., 356 (2013), 301-330.
doi: 10.1007/s00208-012-0849-2. |
[22] |
J. Simon, Compact set in the space $ L^p(0, T; B) $, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
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