This paper focuses on investigating the dynamic behavior of the three-dimensional globally modified Navier–Stokes equations with memory term, based on the general framework of dynamical systems arising from differential equations with delay terms. The method we employed diverges from previous approaches investigating the dynamic behavior of equations with memory term by Dafermos transformations. First, we use the classical Galerkin method to prove the well-posedness of the solutions to the transformation equation. Subsequently, by constructing an appropriate linear mapping, we demonstrated the existence of solutions to the original equation, thereby the semigroup generated by the original equation is established in phase space . Lastly, by showing the existence of bounded absorbing sets for the semigroup and its asymptotic compactness, the existence of global attractors is proved.
AnhC. T.ThanhD. T. P.ToanN. D. (2021). Uniform attractors of 3D Navier–Stokes–Voight equations with memory and singularly oscillating external forces. Evolution Equations and Control Theory, 10, 1–23.
2.
BortolanM. C.CarvalhoA. N.Marín-RubioP.ValeroJ. (2025). Weak global attractor for the 3D Navier–Stokes equations via the globally modified Navier–Stokes equations. Journal of Evolution Equations, 25, 1–29.
3.
CaraballoT.KloedenP. E.RealJ. (2008). Invariant measures and statistical solutions of the globally modified Navier–Stokes equations. Discrete and Continuous Dynamical Systems - Series B, 10, 760–781.
4.
CaraballoT.RealJ.KloedenP. E. (2006). Unique strong solutions and V-attractors of a three-dimensional system of globally modified Navier–Stokes equations. Advanced Nonlinear Studies, 6, 411–436.
5.
Di PlinioF.GiorginiA.PataV.TemamR. (2018). Navier–Stokes–Voigt equations with memory in 3D lacking instantaneous kinematic viscosity. Journal of Nonlinear Science, 28, 653–686.
6.
DongB. Q.SongJ. (2011). Global regularity and asymptotic behavior of modified Navier–Stokes equations with fractional dissipation. Discrete and Continuous Dynamical Systems, 32, 57–79.
7.
FreitasM. M.AraújoG. M.FonsecaJ. S. (2021). Attractors for Navier–Stokes equations with fractional operator in the memory term. Annali dell’Universita’ Di Ferrara. Sezione VII – Scienze Matematiche, 67, 269–284.
8.
GalC. G.Tachim MedjoT. (2013). A Navier–Stokes–Voight model with memory. Mathematical Methods in the Applied Sciences, 36, 2507–2523.
9.
GaldiG. P. (2009). Navier–Stokes equations: A mathematical analysis. In Encyclopedia of complexity and systems science (pp. 5925–5959). Springer.
10.
KloedenP. E.LangaJ.RealJ. (2007). Pullback V-attractors of the 3-dimensional globally modified Navier–Stokes equations. Communications on Pure and Applied Analysis, 6, 937–955.
11.
LiY. R.CaraballoT.WangF. L. (2025). Cardinality and IOD-type continuity of pullback attractors for random nonlocal equations on unbounded domains. Mathematische Annalen, 391, 747–789.
12.
MaQ. F.WangS. H.ZhongC. K. (2002). Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana University Mathematics Journal, 51, 1541–1570.
13.
Marín-RubioP.Márquez-DuránA. M.RealJ. (2013). Asymptotic behavior of solutions for a three-dimensional system of globally modified Navier–Stokes equations with a locally Lipschitz delay term. Nonlinear Analysis, Theory, Methods & Applications, 79, 68–79.
14.
MarionM.TemamR. (2001). Elements of the mathematical theory. In Navier–Stokes equations: Theory and numerical analysis (pp. 509–598). American Mathematical Society.
15.
RobinsonJ. C. (2001). Infinite-dimensional dynamical systems: An introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge University Press.
16.
RomitoM. (2009). The uniqueness of weak solutions of the globally modified Navier–Stokes equations. Advanced Nonlinear Studies, 9, 425–427.
17.
SuK. Q.YangR. (2022). Pullback dynamics and robustness for the 3D Navier–Stokes–Voight equations with memory. Electronic Research Archive, 31, 928–946.
18.
TuanA. N.CaraballoT.TuanN. H. (2022). On the initial value problem for a class of nonlinear biharmonic equations with time-fractional derivative. Proceedings of the Royal Society of Edinburgh Section A, 152, 989–1031.
19.
WangF. L.CaraballoT.LiY. R. (2025). Asymptotic behavior of nonlocal semilinear degenerate heat equations with degenerate memory. Asymptotic Analysis, 141, 135–156.
20.
WangR. H.CaraballoT.TuanN. H. (2023). Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: Theoretical results and applications. Proceedings of the American Mathematical Society, 151, 2449–2458.
21.
WangX. Q.QinY. M. (2021). Three-dimensional Navier–Stokes–Voight equation with a memory and the Brinkman–Forchheimer damping term. Mathematical Methods in the Applied Sciences, 44, 2402–2419.
22.
XuJ. H.CaraballoT.ValeroJ. (2022a). Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion. Journal of Differential Equations, 327, 418–447.
23.
XuJ. H.CaraballoT.ValeroJ. (2022b). Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete and Continuous Dynamical Systems – Series S, 15, 3059–307.
24.
XuJ. H.CaraballoT.ValeroJ. (2024a). Dynamics and Wong–Zakai approximations of stochastic nonlocal PDEs with long time memory. Qualitative Theory of Dynamical Systems, 23, 228.
25.
XuJ. H.CaraballoT.ValeroJ. (2024b). Dynamics and large deviations for fractional stochastic partial differential equations with Lévy noise. SIAM Journal on Mathematical Analysis, 56, 1016–1067.
26.
ZhangY. B.WangX.GaoC. H. (2017). Strong global attractors for nonclassical diffusion equation with fading memory. Advances in Difference Equations, 2017, 163.
27.
ZhaoC. D.CaraballoT. (2019). Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations. Journal of Differential Equations, 266, 7205–7229.
28.
ZhaoC. D.LiY. J.CaraballoT. (2020). Trajectory statistical solutions, Liouville type equations for evolution equations: Abstract results and applications. Journal of Differential Equations, 269, 467–494.
29.
ZhaoC. D.YangL. (2017). Pullback attractors and invariant measures for the non-autonomous globally modified Navier–Stokes equations. Communications in Mathematical Sciences, 15, 1565–1580.
