Global dynamics and asymptotic stability for a nonlocal LANS-$\alpha$-type system

This paper is devoted to the global dynamics of a nonlocal Lagrangian-averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$)-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: $$\mathrm{\parallel }u(t,\cdot ){\mathrm{\parallel }}_{H^3}\lesssim t^{-1\mathrm{/}2},\mathrm{\forall }t>0,$$ 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-$\alpha$-type system.