Landau-Lifshitz Equations

Landau-Lifshitz Equations

Volumn , Issue , 2008, Pages 1-403

  Guo, Boling ^a   Ding, Shijin ^b  

^a INSTITUTE OF APPLIED PHYSICS AND COMPUTATIONAL MATHEMATICS   (China)

^b SOUTH CHINA NORMAL UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85130668523     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/6658     Document Type: Book

References (159)

1. Aïssa, N. and Hamdache, K., Energy decay of solutions to Maxwell’s equations with conductivity and polarization, to appear.
2. Aïssa, N. and Hamdache, K., Asymptotics of time harmonic solutions to a thin ferroelectric model, to appear.
3. Akhiezer, A. I., Yakhtar, V. G. and Peletminskii, S. V., Spin Waves (North-Holland, 1968).
4. Alouges, F. and Soyeur, A., On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal. TMA 18 (1992) 1071-1084.
5. Amann, H., Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rat. Mech. Anal. 92 (1986) 153-192.
6. Ammari, H. and Hamdache, K., Global existence and regularity of solutions to a system of nonlinear Maxwell equations, J. Math. Anal. Appl. 286 (2003) 51-63.
7. Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci. 21 (1998) 823-864.
8. Aubin, T., Nonlinear Analysis on Manifolds, Monge-Ampere Equations (Springer-Verlag, 1982).
9. Aygelman, S. D., Parabolic System (1994).
10. Babin, A. V. and Visik, M. I., Attractors of partial differential equations and estimate of their dimension, Vspekhi Mat. Nauk. 38 (1983) 133-187.
11. Babin, A. V. and Visnik, M. I, Regular attractors of semi-groups and evolution equations, J. Math Pures Appl. 62 (1983) 441-491.
12. Balakrishnan, R., On the inhomogeneous Heisenberg chain, Phys. C. Solid State Phys. 15 (1982) L1305-L1308.
13. Bartels, S., Ko, J. and Prohl, A., Numerical approximation of the Landau-Lifshitz- Gilbert equation and finite time blow-up of weak solutions, preprint, 2005.
14. Bassiouny, E., Ghaleb, A. F. and Maugin, G. A., Thermodynamical formulation for coupled electromechanical hysteresis effects-I. Basic equations, Int. J. Engrg. Sci. 26 (1988) 1279-1295.
15. Bethuel, F., Brezis, H. and Helein, F., Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Vr. P.D.E. 1 (1993) 123-148.
16. Bethuel, F., Brezis, H. and Helein, F., Ginzburg-Landau Vortices (Birkhäuser, 1994).
17. Böttcher, C. J. F., Theory of Electric Polarization (Elsevier, 1952).
18. Carbou, G., Regularity for critical points of a nonlocal energy, Calc. Var. 5 (1997) 409-433.
19. Carbou, G., Thin layers in micromagnetism, Math. Model. Meth. Appl. Sci. 11 (2001) 1529-1546.
20. Carbou, G. and Fabrie, P., Regular solutions for Landau-Lifshitz equation in a bounded domain, Diff. Int. Eqns. 14 (2001) 213-229.
21. Carbou, G. and Fabrie, P., Time average in micromagnetism, J. Diff. Eqns. 147 (1998) 383-409.
22. Carbou, G., Fabrie, P. and Gués, O., Couche limite dans un modéle de ferro-magnétisme, Comm. P.D.E. 27 (2002) 1467-1495.
23. Carbou, G., Fabrie, P. and Gués, O., Boundary layers for Landau-Lifshitz equations, Physica B 343 (2004) 331-336.
24. Carbou, G. and Labbe, S., Stability for static walls in ferromagnetic nanowires, Proc. ENUMATH 2005 (Springer, 2006), pp. 539-546.
25. Chang, K. C., Heat flow and boundary value problem for harmonic map, Anal. Non-Lineaire 6 (1989) 363-395.
26. Chang, K. C., Ding, W. Y. and Ye, R., Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Diff. Geom. 36 (1992) 507-515.
27. Chang, N.-H., Shatah, J. and Uhlenbeck, K., Schrödinger maps, Comm. Pure Appl. Math. 53(5) (2000) 590-602.
28. Chen, Y., The weak solutions to the evolution problems of harmonic maps, Math. Z. 201 (1989) 69-74.
29. Chen, Y., Ding, S. and Guo, B., Partial regularity for two-dimensional Landau- Lifshitz equations, Acta Math. Sinica, New Ser. 14 (1998) 423-432.
30. Chen, Y. and Ding, W. Y., Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990) 567-578.
31. Chen, Y. and Guo, B., Landau-Lifshitz equation from Riemannian surfaces, J. P.D.E. 9 (1996) 313-322.
32. Chen, Y., Li, J. and Lin, F. H., Partial regularity for weak heat flows into spheres, Comm. Pure Appl. Math. 48 (1995) 429-448.
33. Chen, Y. and Lin, F. H., Evolution of harmonic maps with Dirichlet boundary conditions, Comm. Anal. Geom. 1 (1993) 327-346.
34. Chen, Y. and Struwe, M., Existence and partial regularity results for the heat flow of harmonic maps, Math. Z. 201 (1989) 83-103.
35. Chen, Y. and Wang, C., Partial regularity for weak flows into Riemannian homogeneous space, Comm. P.D.E. 21 (1996) 735-761.
36. Coifman, R. and Fefferman, R., Weighted norm inequality for maximal functions and singular integrals, Studia Math. 51 (1974) 241-250.
37. Coifman, R., Lions, P. L., Meyer, Y. and Siemmes, S., Compacite par compensation et espass de Hardy, C. R. Acad Sci. Paris 30 (1989) 945-949.
38. Coron, J., Nonuniqueness for the heat flow of harmonic maps, Ann. Inst. H. Poincaré, Non-Linéairé 7 (1990) 335-344.
39. Coron, J. and Ghidaglia, J., Explosion en temps fini pour le flot des applications harmoniques, (French) [Finite-time blow-up for the heat flow of harmonic mappings] C. R. Acad. Sci. Paris, I Math. 308 (1989) 339-344.
40. Constantin, P., Foias, C. and Temam, R., Attractor representing turbulent flows, Mem. Am. Math. Soc. 53 (1985) 314-380.
41. Cooper, J. and Strauss, W., The initial boundary problem for the Maxwell equations in the presence of a moving boundary, SIAM J. Math. Anal. 16 (1985) 1165-1179.
42. Davi, F., On domain switching in deformable ferroelectrics, seen as continua with microstructure, Z. Angew. Math. Phys. 52 (2001) 966-989.
43. DeBye, P., Polar Molecules (Dover, 1945).
44. DeSimone, A., Kohn, R. V., Müller, S. and Otto, F., A reduced theory for thin film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1408-1460.
45. DeSimone, A., Kohn, R. V., Müller, S. and Otto, F., Magnetic microstructure-a paradigm of multiscale problem, Proc. ICIAM 99 (Edinburgh), eds. J. Ball and J. C. R. Hunt (1999), pp. 175-199.
46. DeSimone, A., Kohn, R. V., Müller, S. and Otto, F., A compactness result in the gradient theory of phase transitions, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 833-844.
47. Ding, S. and Guo, B., Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (2004) 95-117.
48. Ding, S. and Guo, B., Initial-boundary value problem for higher dimensional Landau- Lifshitz systems, Appl. Anal. 83 (2004) 673-697.
49. Ding, S. and Guo, B., Existence of Partially regular weak solutions to Landau- Lifshitz-Maxwell equations, preprint, 2006.
50. Ding, S., Guo, B. and Su, F., Measure-valued solutions to the strongly degenerate compressible Heisenberg chain equations, J. Phys. Math. 40 (1999) 1153-1162.
51. Ding, S., Guo, B. and Su, F., Smooth Solution for one-dimensional inhomogeneous Heisenberg chain equations, Proc. Roy. Soc. Edinburgh 129A (1999) 1171-1184.
52. Ding, S. and Wang, C., Finite time singularity of Landau-Lifshitz-Gilbert equations, Int. Math. Res. Notices, Vol. 2007(4), 2007.
53. Donnat, P. and Rauch, J., Global solvability of the Maxwell-Bloch equations from nonlinear optics, Arch. Rat. Mech. Anal., 136(3) (1996) 291-303.
54. Eidel’man, S. D., Parabolic Systems (North-Holland, 1969).
55. Evans, L. C., Partial regularity for harmonic maps into spheres, Arch. Rat. Mech. Anal. 116 (1991) 101-113.
56. Evans, L. C. and Gariepy, R. F., Measure Theory and Fine Properties of Functions, Studies in Advances Math. (CRC Press, 1992).
57. p spaces of several variables, Acta. Math. 129 (1972) 137-193.
58. Feldman, M., Partial regularity for harmonic maps of evolution into spheres, Comm. P.D.E. 19 (1994) 761-790.
59. Fivez, J., On the continuum limit of a classical compressible Heisenberg Chain, Phys. C: Solid State Phys. 15 (1982) L641-L643.
60. Fogedby, H. C., Theoretical Aspects of Mainly Low Dimensional Magnetic System, Lect. Notes. Phys., Vol. 131 (Springer, 1980).
61. Freire, A., Uniqueness for the harmonic heat flow in two dimensions, Calc. Var. P.D.E. 4 (1996) 761-790.
62. Fröhlich, H., Theory of Dielectrics (Oxford University Press, 1949).
63. Ghidaglia, J. M., Marison, M. and Temam, R., Generalization of the Sobolev-Lieb- Thirring inequalities and applications to the dimension of attractors, Diff. Int. Eqns. 1 (1988) 1-21.
64. Giaquinta, M., Introduction to Regularity Theory for Nonlinear Elliptic Systems (Birkhäuser Verlag, 1993).
65. Gibson, G. and Jeffries, C., Phys. Rev. 429 (1984) 811.
66. Gill, T. L. and Zachary, W. W., Existence and Finite Dimension Ality of Attractors for the Landau-Lifshitz Equations, in Differential Equations in Mathematical Physics, eds. I. Knowles and Y. Saito, Lecture Note in Mathematics (Springer-Verlag, 1985).
67. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (Springer-Verlag, 2001).
68. Gioia, G. and James, R. D., Micromagnetics of very thin films, Proc. Roy. Soc. Lond., Ser. A 453 (1997) 213-223.
69. Greenberg, J. M., MacCamy, R. C. and Coffman, C. V., On the long-time behavior of ferroelectric systems, Physica D 134 (1999), 362-383.
70. Grisvard, P. P., Equations differentiales abstraites, Ann. Scient. EC. Norm. Sup. 4 Serie. T. 2 (1969) 311-395.
71. Guo, B. and Chen, D., On infinite conservation laws of generalized Heisenberg equation, Arch. Math. Sci. 13 (1993) 298-302.
72. Guo, B. and Ding, S., Initial-boundary value problem for the Landau-Lifshitz system(I)-existence and partial regularity, Progr. Nat. Sci. 8 (1998) 11-23.
73. Guo, B. and Ding, S., Initial-boundary value problem for the Landau-Lifshitz(II)-uniqueness, Progr. Nat. Sci. 8 (1998) 147-151.
74. Guo, B. and Ding, S., Neumann problem for the Landau-Lifshitz-Maxwell system in two dimensions, Chin. Ann. Math., Ser. B 22 (2001) 529-540.
75. Guo, B. and Ding, S., Initial boundary value problem for the unsaturated Landau- Lifshitz system of ferromagnetic spin chain, Chin. Ann. Math., Ser. B 21 (2000) 389-402.
76. Guo, B. and Ding, S., Initial boundary value problem for the Landau-Lifshitz system with applied fields, Proc. on Nonlinear Partial Differential Equations and Applications (World Scientific, 1998).
77. Guo, B. and Hong, M., The Landau-Lifshitz equations of the ferromagnetic spin chain and harmonic maps, Calc. Var. 1 (1993) 311-334.
78. Guo, B., Hong, M. C. and Su, F., The Attractors for the Landau-Lifshitz Equation of the Ferromagnetic Spin Chain on Compact Manifold, Beijing Math., Vol. 2 (1996), pp. 40-75; AMS/IP Stud. Adv. Math. 3 (Amer. Math. Soc., 1997), pp. 213-227.
79. Guo, B. and Lu, B., Stability for steady solution of Landau-Lifshitz equation, preprint.
80. Guo, B. and Su, F., Global weak solution for the Landau-Lifshitz-Maxwell equation in three space dimensions, J. Math. Anal. Appl. 211 (1997) 326-346.
81. Guo, B. and Su, F., The global smooth solution for Landau-Lifshitz-Maxwell equations without dissipation, J. Partial Diff. Eqns. 11(3) (1998) 193-208.
82. Guo, B. and Su, F., The global solution for Landau-Lifshitz-Maxwell equations, J. Partial Diff. Eqns. 14(2) (2001) 133-148.
83. Guo, B. and Su, F., The attractors for Landau-Lifshitz-Maxwell equations, J. Partial Diff. Eqns. 13(4) (2000) 320-340.
84. Guo, B. and Su, F., Global weak solution for Landau-Lifshitz-Maxwell equations on Riemannian manifold, preprint.
85. Guo, B. and Wang, B., Global existence of the solution for the Landau-Lifshitz equation of the ferromagnetic spin chain, Math. Sin. 17 (1997) 429-436.
86. Guo, B. and Wang, B., Approximation to the global attractor for the Landau-Lifshitz equation of the forromagnetic spin chain, Beijing Math. 1 (1995) 160-175.
87. Guo, B. and Wang, Y., Generalized Landau-Lifshitz systems of the ferromagnetic spin chain and harmonic maps, Science in China, Ser. A 39 (1996) 1242-1287.
88. Hamdache, K. and Ngningone, I., Time harmonic solutions to a model of ferroelectric materials, Appl. Anal. (2005) 1-16.
89. Harpes, P., Partial compactness for the 2-D Landau-Lifshitz flow, Electronic J. Diff. Eqns. 90 (2004) 1-24.
90. Helein, F., Regularite des Applications Faiblerment Harmoniques Entre Une Surface et Une Sphere, Comptes Rendus Acad. Sci., Serie I T311 (1990) 519-524.
91. Hungerbuhler, N., private communication.
92. Iwaniec, T. and Martin, G., Quasiregular mappings in even dimension, Acta Math.
93. Jackson, J. D., Classical Electrodynamics, 3rd edn., Chinese version (Higher Education Press, 2004).
94. Jochmann, F., Long Time asymptotics of solutions to the anharmonic oscillator model from nonlinear optics, SIAM J. Math. Anal. 32 (2000) 887-915.
95. Joly, J. L., Métivier, G. and Rauch, J., Global solvability of the anharmonic oscillator model from nonlinear optics, SIAM J. Math. Anal. 27 (1996) 905-913.
96. Kamlah, M. and Tsakmakis, C., Phenomenological modeling of the non-linear electromechanical coupling in ferroelectrics, Int. J. Solids Str. 36 (1999) 669-695.
97. Klaina, S., Global existence of nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980) 43-101.
98. Kohn, R. V. and Slastikov, V. V., Another thin film limit of micromagnetics, Arch. Rat. Mech. Anal. 178 (2005) 227-245.
99. Kurzke, M., Melcher, C. and Moser, R., Domain walls and vortices in the thin ferromagnetic films, Physica B 343 (2004) 331-336.
100. Kurzke, M., Melcher, C. and Moser, R., Domain walls and vortices in thin ferromagnetic films, MPI Leipzig preprint, No. 45 (2006).
101. Ladyzhenskaja, O., Solonnikov, V. A. and Ural’ceva, N. N., Linear and Quasilinear Equations of Parabolic Type, AMS Transl. Math. Monogr., Vol. 23 (Amer. Math. Soc., 1968).
102. Landau, L. D. and Lifshitz, E. M., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sovietunion. 8 (1935) 153-169.
103. Landau, L. D. and Lifshitz, E. M., Electrodynamique des milieux continuous, Cours de Physique Theorique, Vol. VIII (MIR, 1969).
104. Landau, L. D. and Lifshitz, E. M., Electrodynamics of Continuous Media (Pergamon Press, 1960).
105. Lakshmanan, M., Ruijgrok, T. W. and Thompson, C. J., On the dynamics of a continuous spin systems, Phys. A 84A (1976) 577-590.
106. Lakshmannan, M. and Nakamura, K., Landau-Lifshitz equation of ferromagnetism: Exact treatment of the Gilbert damping, Phys. Rev. Lett. 53 (1984) 2497-2499.
107. Lakshmanan, M. and Daniel, M., Soliton damping and energy loss in the classical continuum Heisenberg spin chain, Phys. Rev. B 24 (1981) 6750-6754.
108. Lin, J. and Ding, S., Smooth solution to the one dimensional Inhomogeneous non-automorphic Landau-Lifshitz equation, Proc. Roy. Soc. Lond., Ser. A 462 (2006) 2397-2413.
109. Liu, X., Partial regularity for Landau-Lifshitz system of ferromagnetic spin chain, Calc. Var. 20 (2004) 153-173.
110. Magyari, E., Solitary waves along the compressible Heisenberg chain, Phys. C: Solid State Phys. 15 (1982) L1159-L1163.
111. Malek, J., Necas, J., Rokyta, M. and Ruzicka, M., Weak and Measure-Valued Solutions to Evolutionary PDES (Chapman & Hall, 1996).
112. Melcher, C., Domain wall motion in ferromagnetic films, Phys. D. 192 (2004) 249- 264.
113. 3, Comm. P.D.E. 30 (2005) 567-587.
114. Michelitsch, T. and Kreher, W. S., A simple model for the nonlinear material behavior of ferroelectrics, Int. Acta Mater. 46 (1998) 5085-5094.
115. Moser, R., Partial regularity for the Landau-Lifshitz equation in small dimensions, MPI (Leipzig) preprint, 2002.
116. Moser, R., Ginzburg-Landau vortices in thin ferromagnetic films, ARMX Appl. Math. Res. Express 1 (2003) 1-32.
117. Moser, R., Boundary vortices for thin ferromagnetic films, Arch. Rat. Mech. Anal. 174 (2004) 267-300.
118. Moser, R., Partial regularity for Landau-Lifshitz equations, Preprint Series, Max-Planck-Institute for Mathematics in the Sciences 26.
119. Moser, R., Partial Regularity for Harmonic Maps and Related Problems (World Scientific, 2005).
120. Moser, R., Moving boundary vortices for a thin film limit in micromagnetics, Comm. Pure Appl. Math. 58 (2005) 701-721.
121. Nahamura, C. K., Ohta, S. and Kawasahi, K., Chaotic states on ferromagnets in strong parallel pumping fields, J. Phys. C 15 (1982) 143-148.
122. Nakamura, K. and Sasada, T., Soliton and wave trains in ferromagnets, Phys. Lett. 48A (1974) 321-322.
123. Pistella, F. and Valente, V. Numerical study of the appearance of singularities in ferromagnets, Adv. Math. Sci. Appl. 12 (2002) 803-816.
124. Porsezian, K. and Lakshmanan, M., On the dynamics of the radially symmetric Heisenberg ferromagnetic spin system, J. Math. Phys. 32 (1991) 2923-2928.
125. 2, C. R. Acad. Sci. Paris 314 (1992) 719-723.
126. Riviére, T., Flot des applications harmoniques en dimension deux (1993).
127. Schoen, R. and Yau, S. T., Differential Geom. (Academic Press, 1998).
128. Semmes, S., A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. P.D.E. 19 (1994) 277-319.
129. Siman, J., Nonhomogeneous viscous incompressible fluids existence of vecocity, density and pressure, SIAM J. Math. Anal. 21 (1990) 1039-1117.
130. Su, F. Q. and Guo, B., The global smooth solution for Landau-Lifshitz-Maxwell equation without dissipation, J. P.D.E. 11 (1998) 193-208.
131. Suhl, H., The theory of ferromagnetic resonance at high signal powers, J. Phys. Chem. Solids (1957) 209-227.
132. Sulem, P. L., Sulem, C. and Bardos, C., On the continuous limit for a system of classical spin, Comm. Math. Phys. 107 (1986) 431-454.
133. Struwe, M., On the evolution of harmonic maps on Riemannian surface, Comm. Math. Helv. 60 (1985) 558-581.
134. Struwe, M., On the evolution of harmonic maps in higher dimensions, J. Diff. Geom. 28 (1988) 485-502.
135. Struwe, M., Geometric Evolution Problems, in Nonlinear Partial Differential Equations in Differential Geometry (Park City, UT, 1992), IAS/Park City Math. Ser., 2 (Amer. Math. Soc., 1996), pp. 257-339.
136. Stein, E. M., Singular Integrals and Differentiability Properties of Functions (Princeton University Press, 1970).
137. Takhtalian, L. A., Integration of the continuous Heisenberg spin chain through the inverse scattering method, Phys. Lett. 64A (1977) 235.
138. Temam, R., Infinite Dimensional Dynamical in Mechanics and Physics (Springer-Verlag, 1998).
139. Tjon, J. and Wright, J., Soliton in the continuous Heisenberg chain, Phys. Rev. B 15 (1977) 3470-3476.
140. Visintin, A., On Landau-Lifshitz equations for ferromagnetism, Jpn. J. Appl. Math. 2 (1985) 69-84.
141. Visintin, A., Maxwell equations with vector hysteresis, Arch. Rat. Mech. Anal., DOI:10.1007/s00205-004-0333-6.
142. Wang, C. Y., On moving Ginzburg-Landau vortices, Comm. Anal. Geom. 12 (2004) 1185-1199.
143. Wang, C. Y., On Landau-Lifshitz equation in dimensions at most four, to appear, Indiana Univ. Math. J.
144. Wente, H., An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969) 318-344.
145. White, B., Infima of energy functionals in homotopy classes of mappings, J. Diff. Geom. 23 (1986) 127-142.
146. Zakharov, V. E. and Tekhtajan, L. A., Equivalence of nonlinear Schrödinger equation and Heisenberg ferromagnet, Theor. Mat. Phys. 38 (1979) 17.
147. Zhang, X. Y. and Suhl, H., Spin wave-related period doublings and chaos under transverse pumping, Phys. Rev. A 32 (1985) 2530-2533.
148. Zhong W., Physics of Ferroelectric (in Chinese) (Science Press, 2000).
149. Zhou, Y. L., Finite difference solutions of the nonlinear mutual boundary problems for the ferromagnetic chain systems, J. Comput. Math. 2 (1984) 263-271.
150. Zhou, Y., Sun, H. and Guo, B., Existence of weak solution for boundary problems of systems of ferromagnetic chain, Sci. Sin. A 27 (1981) 779-811.
151. xx + f(x, t, Z), Proc. DD-3 Symp. (1982) 713-732.
152. Zhou, Y., Sun, H. and Guo, B., Finite difference solutions of the boundary problems for systems of ferromagnetic chain, J. Comp. Math. 1 (1983) 294-302.
153. Zhou, Y., Sun, H. and Guo, B., Existence of weak solution for boundary problems of ferromgnetic chain, Sci. Sin. A 27 (1984) 799-811.
154. Zhou, Y., Sun, H. and Guo, B., The weak solution of homogeneous boundary value problem for the system of ferromagnetic chain with several variables, Sci. Sin. 4(A) (1986) 337-349.
155. Zhou, Y., Sun, H. and Guo, B., Some boundary problems of the spin system and the system of ferro magnetic chain I: Nonlinear boundary problems, Acta Math. Sci. 6 (1986) 321-337.
156. Zhou, Y., Sun, H. and Guo, B., Some boundary problems of the spin system and the system of ferromagnetic chain II: Mixed problems and others, Acta Math. Sci. 7 (1987) 121-132.
157. Zhou, Y., Sun, H. and Guo, B., Weak solution systems of ferromagnetic chain with several variables, Science in China A30 (1987) 1251-1266.
158. Zhou, Y., Guo, B. and Tan, S., Existence and uniqueness of smooth solution for system of ferromagnetic chain, Science in China, Ser. A 34 (1991) 257-266.
159. Zhou, Y., Sun, H. and Guo, B., Geometrical extensions for systems of ferromagnetic chain, Science in China, Ser. A 36 (1993) 927-939; 36 (1993) 1422-1434.