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## Smooth Solution to the 1-Dimensional Spin Equations

### Citations

18 |
Existence and uniqueness of smooth solution of system of ferromagnetic chain
- Zhou, Guo, et al.
- 1991
(Show Context)
Citation Context ...112k1 = k222k2 . 2 Existence of Local Smooth Solutions To get the existence of local smooth solution of (1.1)–(1.2), we apply the difference method. We need the following well-known lemmas: Lemma 2.1 =-=[4]-=- Let q, r be real numbers and j,m be integers such that 1 ≤ q, r ≤ ∞, 0 ≤ j < m. If u ∈ Wm,r(Ω) ∩ Lq(Ω), then ‖Dju‖p ≤ C‖u‖1−αq ‖Dmu‖αr , Smooth Solution to the 1-Dimensional Spin Equations of Antifer... |

2 | Theoretical Aspects of Mainly Low Dimensional Magnetic Systems, - Fogedby - 1980 |

2 | On the inhomogeneous Heisenberg chain - Balakrishnan - 1982 |

1 |
S.: Spin Waves and Ferromagnetic Chain Equations, Zhejiang Sci
- Guo, Ding
(Show Context)
Citation Context ...erature, known as the Néel temperature. Without taking dissipation into account, the equations of motion for the magnetizations m1 and m2 of the two magnetic sublattics read as follows (see [1] or =-=[3]-=-):{ mt = 2k1 m×∆m + k11 m×∆n, nt = 2k2n×∆n + k22n×∆m. The equations of motion for the spin wave of antiferromagnets take the following form (see also [1] or [3]): { mt = ∆m + 2k1 m×∆m + ... |

1 |
F.: Smooth solution for the 1D inhomogeneous Heisenberg chain equations
- Ding, Guo, et al.
- 1171
(Show Context)
Citation Context ...3 The first author is supported by the Natural Science Foundation of China (No. 19971030) and the Natural Science Foundation of Guangdong (No. 000671, No. 031495) 2 Ding S. J. and Guo B. L. equations =-=[5]-=- and the compressible Heisenberg chain equations [6]. These models were introduced in [1, 7–9]. In antiferromagnets, the mean atomic magnetic moments compensate each other within each unit cell (in ze... |

1 |
F.: Measure-valued solution to the compressible Heisenberg chain equations
- Ding, Guo, et al.
- 1999
(Show Context)
Citation Context ...ce Foundation of China (No. 19971030) and the Natural Science Foundation of Guangdong (No. 000671, No. 031495) 2 Ding S. J. and Guo B. L. equations [5] and the compressible Heisenberg chain equations =-=[6]-=-. These models were introduced in [1, 7–9]. In antiferromagnets, the mean atomic magnetic moments compensate each other within each unit cell (in zero external magnetic field). In other words, an anti... |

1 | On the continuum limit of a classical compressible Heisenberg chain - Fivez - 1982 |

1 | F.: Solitary waves along the compressible Heisenberg chain - Magyari - 1982 |

1 |
B.: Weak solutions to the spin equations of antiferromagnets
- Ding, Guo
- 2000
(Show Context)
Citation Context ...k11 m×∆n, nt = ∆n + 2k2n×∆n + k22n×∆m, where k1, k2, k11, k22 are constants. Note that in the above equations, m,n are both threedimensional vectors. Using the Galerkin method, we proved in =-=[10]-=- that the above equations with periodic initial values in d-dimensions admit at least one global weak solution. Moreover, we also gave some regularity results for the weak solutions in [10]. In this p... |

1 |
L.: Interpolation formulas of intermediate quotients for discrete functions with several indices
- Zhou
- 1984
(Show Context)
Citation Context ...ju‖p ≤ C‖u‖1−αq ‖Dmu‖αr , Smooth Solution to the 1-Dimensional Spin Equations of Antiferromagnets 3 where ‖ · ‖p = ‖ · ‖Lp(Ω), p ≥ 1, jm ≤ α ≤ 1 and 1 p − j = 1− α q + α (1 r −m ) , Ω ⊂ R1. Lemma 2.2 =-=[11]-=- Let p be a real number and j,m be integers such that 2 ≤ p ≤ ∞, 0 ≤ j < m. Then ‖δjuh‖p ≤ C‖uh‖1−α2 ( ‖δmuh‖2 + ‖uh‖2(2D)m )α , where uh = {uj = u(xj) | j = 0, 1, 2, . . . , J}, xj = jh, h = 2D/J , α... |