Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 3

Almost continuity of a pullback random attractor for the stochastic $g$-Navier–Stokes equation

Pages: 231 – 256

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n3.a4


Yangrong Li (School of Mathematics and Statistics, Southwest University, Chongqing, China)

Shuang Yang (School of Mathematics and Statistics, Southwest University, Chongqing, China)


A pullback random attractor for a cocycle is a family of compact invariant attracting random sets $A(t, \theta_s \cdot)$, where $(t, s)$ is a point of the Euclid plane and $\theta$ is a group of measure-preserving transformations on a probability space. Under three conditions including the union closedness of the universe, the time-sample compactness of the PRA and the joint continuity of the cocycle, we prove that the map $(t, s) \to A(t, \theta_s \cdot)$ is continuous at all points of a residual diagonal-closed subset of the Euclid plane and full pre-continuous with respect to the Hausdorff metric. Applying to the non-autonomous stochastic $g$‑Navier–Stokes equation, we show the sample-continuity and local-uniform asymptotic compactness of the cocycle, which lead to the existence, residual continuity and pre-continuity of a PRA.


stochastic $g$-Navier–Stokes equation, random attractor, residual continuity, pre-continuity, pullback attractor

2010 Mathematics Subject Classification

Primary 37L55. Secondary 35B41, 60H15.

This work is supported by National Natural Science Foundation of China grants 11571283.

Received 23 June 2020

Published 22 July 2021