The existence of a random attractor for the three dimensional damped Navier–Stokes equations with additive noise

ABSTRACT

This article is concerned with the asymptotical behavior of solutions for the three-dimensional damped Navier–Stokes equations with additive noise. Due to the shortage of the existence proof of the existence of random absorbing sets in a more regular phase space, we cannot obtain some kind of compactness of the cocycle associated with the three-dimensional damped Navier–Stokes equations with additive noise by the Sobolev compactness embedding theorem. In this paper, we prove the existence of a random attractor for the three-dimensional damped Navier–Stokes equations with additive noise by verifying the pullback flattening property.

KEYWORDS:

• Random attractor
• damped Navier–Stokes equations
• pullback flattening property
• additive noise

MATHEMATICS SUBJECT CLASSIFICATION:

• 35Q35
• 37L55
• 37N10

Funding

This work was supported by the National Science Foundation of China Grant (11401459) and the Natural Science Foundation of Shaanxi Province (2015JM1010).