Invariant Measure of Stochastic 3D Globally Modified Navier–Stokes Equations with Infinite Varying Delay

• Published in: Bulletin of the Malaysian Mathematical Sciences Society Vol. 48; no. 5
• Main Authors: Zhao, Wenqiang, Liu, Hao, Liu, Xia
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.09.2025; Springer Nature B.V
• Subjects: Applications of Mathematics; Asymptotic methods; Asymptotic properties; Energy methods; Fluid flow; Invariants; Mathematics; Mathematics and Statistics; Navier-Stokes equations; White noise; 37L55; Energy method; Invariant Borel probability measure; Pullback random attractor; Non-autonomous 3D globally modified Navier–Stokes equations; 35B41; 37L40; Infinite varying delay; 37L30; 60H15
• ISSN: 0126-6705; 2180-4206
• DOI: 10.1007/s40840-025-01893-7

This paper is concerned with the asymptotic dynamics of a stochastic 3 D globally modified Navier–Stokes equations with additive white noise and infinite varying delay. Based on the technically established pullback random absorbing set, we first show the existence and uniqueness of pullback random attractor for such equations. The asymptotic compactness of solutions is verified by mainly employing an energy method over the time compact interval, as well as a limit argument at the negative infinite part. Then by proving the joint continuity of solutions with respect to the initial time and initial data, we eventually construct a family of invariant Borel probability measures supported by the derived pullback random attractor.