Well-posedness of the Stochastic Landau--Lifshitz--Baryakhtar Equation in βd
Abstract
This paper studies the Cauchy problem for the stochastic Landau--Lifshitz--Baryakhtar equation on βd, d=1,2,3, subject to Stratonovich Gaussian perturbations. We establish a low-regularity well-posedness theory for initial data in π2, β1, and β2. For π2 initial data, we prove the existence and pathwise uniqueness of global very weak solutions in dimensions one and two, while in dimension three we construct martingale very weak solutions on every finite time interval. For arbitrary β1 initial data, we prove the existence and uniqueness of global pathwise weak solutions in all dimensions dβ€3. Furthermore, for β2 initial data, we establish the existence and uniqueness of global pathwise strong solutions in the same range of dimensions. The analysis is based on a frequency-truncation approximation scheme combined with stochastic compactness arguments in the whole-space setting. The key step in global continuation is to justify the stochastic effective-field balance at β1 regularity, where the drift is available only in a negative Sobolev space. Spatial mollification and passage to the limit yield the energy identity for local weak solutions. A coercive modification then gives global energy moments and excludes finite-time blow-up.
Related
- The stochastic Landau-Lifshitz-Baryakhtar equation in critical spaces
- Small-Time Asymptotic Behavior of the Stochastic Landau--Lifshitz--Baryakhtar Equation
- Beyond the reduced Landau--Lifshitz--Bloch equation: Singular potential, global well-posedness, and micromagnetic limits
- Rate of convergence of a fully discrete structure-preserving midpoint scheme for the stochastic Landau--Lifshitz--Gilbert equation
- Uniform Large Deviations, Barrier Identity and the NΓ©el--Arrhenius Law for the One-Dimensional Stochastic Landau--Lifshitz--Gilbert Equation
BibTeX
@misc{liu2026wellposedness,
title = {{Well-posedness of the Stochastic Landau--Lifshitz--Baryakhtar Equation in $\mathbbR^d$}},
author = {Bin Liu and Fan Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38888},
eprint = {2609.38888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38888},
}
RIS
TY - UNPB
TI - Well-posedness of the Stochastic Landau--Lifshitz--Baryakhtar Equation in β^d
AU - Bin Liu
AU - Fan Xu
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38888
UR - https://arxiv.org/abs/2609.38888
AB - This paper studies the Cauchy problem for the stochastic Landau--Lifshitz--Baryakhtar equation on β^d, d=1,2,3, subject to Stratonovich Gaussian perturbations. We establish a low-regularity well-posedness theory for initial data in π^2, β^1, and β^2. For π^2 initial data, we prove the existence and pathwise uniqueness of global very weak solutions in dimensions one and two, while in dimension three we construct martingale very weak solutions on every finite time interval. For arbitrary β^1 initial data, we prove the existence and uniqueness of global pathwise weak solutions in all dimensions dβ€3. Furthermore, for β^2 initial data, we establish the existence and uniqueness of global pathwise strong solutions in the same range of dimensions. The analysis is based on a frequency-truncation approximation scheme combined with stochastic compactness arguments in the whole-space setting. The key step in global continuation is to justify the stochastic effective-field balance at β^1 regularity, where the drift is available only in a negative Sobolev space. Spatial mollification and passage to the limit yield the energy identity for local weak solutions. A coercive modification then gives global energy moments and excludes finite-time blow-up.
ER -