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

Authors

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

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

Abstract

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.

Keywords

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

2010 Mathematics Subject Classification

Primary 37L55. Secondary 35B41, 60H15.

Received 23 June 2020

Published 22 July 2021