Mathematical study of a new Navier-Stokes-alpha model with nonlinear filter equation - Regularity theory and refined time asymptotics

Abstract: This article is devoted to the mathematical analysis of a new Navier--Stokes-$α$ model involving a nonlinear filter equation. The resulting model is governed by a doubly nonlinear parabolic--elliptic coupled system. In our previous work [M.~F.~Cortez and O.~Jarrín, \emph{Mathematical study of a new Navier--Stokes-alpha model with nonlinear filter equation -- Part I}, J. Math. Fluid Mech. 28, no.~12 (2026)], we established the global well-posedness of weak Leray-type solutions and the existence of a global attractor. In the present work, under natural assumptions on $A(\cdot)$, we investigate several additional properties of this model, with particular emphasis on higher-order regularity and its consequences for the long-time dynamics. The regularity analysis constitutes a central and delicate issue, since the nonlinear structure of the elliptic filter and its coupling with the evolution equation preclude a direct application of the standard regularity arguments available for linearly filtered Navier--Stokes-$α$ models. Overcoming this difficulty requires new higher-order estimates for the nonlinear elliptic filter equation, which may also be of independent interest. Among the results obtained, two constitute the main contributions of the article. First, we establish uniform higher-order regularity for the global attractor. This result is then used as a fundamental ingredient in proving an exact determining-modes property for complete trajectories on the attractor: if the projections of two complete trajectories onto a sufficiently large finite-dimensional space of Stokes modes coincide at every time, then the two trajectories coincide identically.