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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2024; 61(2): 227-253

Online first article February 23, 2024      Printed March 1, 2024

https://doi.org/10.4134/JKMS.j220380

Copyright © The Korean Mathematical Society.

Asymptotic behavior of solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays

Cung The Anh, Vu Manh Toi, Phan Thi Tuyet

Hanoi National University of Education; Thuyloi University; Electric Power University

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This paper studies the existence of weak solutions and the stability of stationary solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays in the phase space $BCL_{-\infty}(H)$. We first prove the existence and uniqueness of weak solutions by using the classical technique of Galerkin approximations. Then we study stability properties of stationary solutions by using several approach methods. In the case of proportional delays, some sufficient conditions ensuring the polynomial stability in both mean square and almost sure senses will be provided.

Keywords: Stochastic globally modified Navier-Stokes equations, unbounded delays, stability, stationary solutions, polynomial stability

MSC numbers: Primary 60H15, 35Q35, 60H30, 35B35

Supported by: This work is financially supported by the Vietnam Ministry of Education and Training under grant number B2023-SPH-13.

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