Beyond the reduced Landau--Lifshitz--Bloch equation: Singular potential, global well-posedness, and micromagnetic limits
Abstract
We study the full nonlinear mean-field Landau--Lifshitz--Bloch equation in micromagnetics arising from Garanin's spin-polarisation model [D. Garanin. Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets. Phys. Rev. B, 55 (1997)] prior to the usual near-equilibrium reduction. The equation combines a singular inverse-Langevin potential with a mobility whose longitudinal component degenerates at saturation, together with exchange, anisotropy, and the nonlocal whole-space stray field. The natural state space is the closed unit ball |m|≤ 1. For any H1 initial data satisfying |m0|≤1, including saturated data of infinite free energy, we construct global weak solutions and derive instantaneous separation from saturation |m|=1. This restores uniform parabolicity and yields the thermodynamic formulation and exact energy identity for positive times. We further establish a quantitative low-temperature limit to the Landau--Lifshitz--Gilbert equation and prove convergence of global trajectories to a single equilibrium. A spectral Curie-type threshold characterises the stability of the paramagnetic state and yields exponential relaxation above the threshold. Finally, for uniformly separated data we prove strong well-posedness, including global strong solutions in one- and two-dimensional domains, and exponential stability of non-degenerate equilibria, with a small-data global theory in three dimensions.
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BibTeX
@misc{soenjaya2026beyond,
title = {{Beyond the reduced Landau--Lifshitz--Bloch equation: Singular potential, global well-posedness, and micromagnetic limits}},
author = {Agus L. Soenjaya},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00365},
eprint = {2611.00365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00365},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Beyond the reduced Landau--Lifshitz--Bloch equation: Singular potential, global well-posedness, and micromagnetic limits
AU - Agus L. Soenjaya
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00365
UR - https://arxiv.org/abs/2611.00365
AB - We study the full nonlinear mean-field Landau--Lifshitz--Bloch equation in micromagnetics arising from Garanin's spin-polarisation model [D. Garanin. Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets. Phys. Rev. B, 55 (1997)] prior to the usual near-equilibrium reduction. The equation combines a singular inverse-Langevin potential with a mobility whose longitudinal component degenerates at saturation, together with exchange, anisotropy, and the nonlocal whole-space stray field. The natural state space is the closed unit ball |m|≤ 1. For any H^1 initial data satisfying |m_0|≤1, including saturated data of infinite free energy, we construct global weak solutions and derive instantaneous separation from saturation |m|=1. This restores uniform parabolicity and yields the thermodynamic formulation and exact energy identity for positive times. We further establish a quantitative low-temperature limit to the Landau--Lifshitz--Gilbert equation and prove convergence of global trajectories to a single equilibrium. A spectral Curie-type threshold characterises the stability of the paramagnetic state and yields exponential relaxation above the threshold. Finally, for uniformly separated data we prove strong well-posedness, including global strong solutions in one- and two-dimensional domains, and exponential stability of non-degenerate equilibria, with a small-data global theory in three dimensions.
ER -