非局部LANS-$\alpha$型系统的全局动力学与渐近稳定性
Global dynamics and asymptotic stability for a nonlocal LANS-$α$-type system
- Guangdong University of Technology（广东工业大学）
- Huazhong University of Science and Technology（华中科技大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究非局部LANS-$\alpha$型系统在无外力周期情形下的全局动力学，证明全局适定性、耗散半群及单点全局吸引子，并给出$H^3$范数以$t^{-1/2}$衰减的代数渐近稳定性估计。
AI中文摘要：
本文致力于研究一维、二维和三维空间中非局部拉格朗日平均纳维-斯托克斯-$\alpha$（LANS-$\alpha$）型系统（也称为粘性欧拉-庞加莱系统）的全局动力学。在无外力、周期边界且均值为零的设定下，我们建立了初值问题的全局适定性，并证明相应的解半群是耗散的，且拥有一个由单点集$\{0\}$组成的紧全局吸引子。此外，我们获得了如下全局时间代数渐近稳定性估计：\\[ \\|u(t,\cdot)\\|_{H^3}\lesssim t^{-1/2},\qquad \forall\\, t>0, \\] 该估计由高阶能量不等式结合一致Gronwall型论证推导得出。我们的结果为该非局部LANS-$\alpha$型系统无外力周期动力学的长时间行为提供了完整描述。
英文摘要：
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: \[ \|u(t,\cdot)\|_{H^3}\lesssim t^{-1/2},\qquad \forall\, t>0, \] 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.
