Uniform higher-order regularity of the global attractor without an a priori filtered-trajectory bound

Establish a uniform higher-order regularity bound for trajectories in the global attractor of the three-dimensional Navier–Stokes-alpha model with nonlinear filter equation under the regularity assumptions on the stationary external force, indicator function, and convolution kernel alone, without assuming the auxiliary bound on the filtered trajectories in $W^{m,infty}$.

Background

The paper proves an $H^{m+1}$ uniform bound for complete trajectories on the global attractor only after imposing the additional condition $\Lambda(\alpha,m)<\nu$, where $\Lambda(\alpha,m)$ is a uniform-in-time $W^{m,\infty}$ bound for the filtered trajectories. This condition ensures that the higher-order energy inequality has a dissipative coefficient.

The authors explicitly state that obtaining the same uniform higher-order estimate from the regularity assumptions on the force, indicator function, and convolution kernel alone is unresolved. Removing the auxiliary smallness or boundedness assumption would yield a stronger regularity theorem for the global attractor.