Existence and regularity of random attractors for g -Navier–Stokes equations driven by nonlinear colored noise

This paper investigates the existence and regularity of pullback random attractors for a class of non-autonomous stochastic g-Navier–Stokes equations driven by nonlinear colored noise. We first establish the existence and uniqueness of pullback random attractors in the g-weighted space Hg. We further prove the regularity of the attractor in the strong topology of Vg, showing that it is a bi-spatial attractor which is compact in Hg while attracting bounded sets in Vg. The main difficulty in proving pullback asymptotic compactness in Hg is overcome by deriving uniform estimates in both Hg and Vg and leveraging the compact embedding Vg ↪ Hg. To establish compactness in the stronger space Vg, we employ a spectral decomposition method. This yields the flattening property of solutions and thereby verifies the asymptotic compactness in Vg.