Global dynamics and asymptotic stability for a nonlocal LANS-α-type system
Abstract
This paper is devoted to the global dynamics of a nonlocal Lagrangian-averaged Navier-Stokes-α (LANS-α)-type system (also known as the viscous Euler-Poincaré system) in one, two, and three spatial dimensions. In the unforced periodic setting with zero mean, we establish global well-posedness for the initial-value problem and prove that the corresponding solution semigroup is dissipative and possesses a compact global attractor consisting of the singleton {0}. Furthermore, we obtain the following global-in-time algebraic asymptotic stability estimate: |u(t,·)|H3≲ t−1/2, ∀ t>0, which is derived from higher-order energy inequalities combined with a uniform Gronwall-type argument. Our results provide a complete description of the long-time behavior of the unforced periodic dynamics for this nonlocal LANS-α-type system.
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BibTeX
@misc{guan2026global,
title = {{Global dynamics and asymptotic stability for a nonlocal LANS-$α$-type system}},
author = {Chunxia Guan and Kai Yan and Qianyuan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25661},
eprint = {2609.25661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25661},
}
RIS
TY - UNPB
TI - Global dynamics and asymptotic stability for a nonlocal LANS-α-type system
AU - Chunxia Guan
AU - Kai Yan
AU - Qianyuan Zhang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25661
UR - https://arxiv.org/abs/2609.25661
AB - This paper is devoted to the global dynamics of a nonlocal Lagrangian-averaged Navier-Stokes-α (LANS-α)-type system (also known as the viscous Euler-Poincaré system) in one, two, and three spatial dimensions. In the unforced periodic setting with zero mean, we establish global well-posedness for the initial-value problem and prove that the corresponding solution semigroup is dissipative and possesses a compact global attractor consisting of the singleton {0}. Furthermore, we obtain the following global-in-time algebraic asymptotic stability estimate: |u(t,·)|_H^3≲ t^−1/2, ∀ t>0, which is derived from higher-order energy inequalities combined with a uniform Gronwall-type argument. Our results provide a complete description of the long-time behavior of the unforced periodic dynamics for this nonlocal LANS-α-type system.
ER -