Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-25T14:12:35.175Z Has data issue: false hasContentIssue false
  • English
  • Français

3D Navier–Stokes–Voigt equations with damping and double delays on unbounded domains: Well-posedness, pullback attractors, and limit measures

Part of: Incompressible viscous fluids Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  18 March 2025

Zhengwang Tao ,
Dandan Yang  and
Yunfei Lv
Zhengwang Tao
Affiliation:
School of Mathematical Sciences, Tiangong University, No. 399 Binshui West Road, Xiqing District, Tianjin, 300387, China ([email protected])
Dandan Yang
Affiliation:
School of Mathematical Sciences, Tianjin Normal University, No. 393 Binshui West Road, Xiqing District, Tianjin, 300387, China ([email protected])
Yunfei Lv*
Affiliation:
School of Mathematical Sciences, Tiangong University, Tianjin, China ([email protected]) (corresponding author)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

This article is dedicated to investigating limit behaviours of invariant measures with respect to delay and system parameters of 3D Navier–Stokes–Voigt equations. Firstly, the well-posedness of such a system is obtained on arbitrary open sets that satisfy the Poincaré inequality, and then a unique minimal pullback attractor is attained by using the energy equation method and asymptotic compactness property. Furthermore, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors. Specifically, when the external forcing terms are periodic in time, the periodic invariant measure can be obtained. Finally, as the delay approaches zero and system parameters tend to some numbers, the limit of the invariant measure sequences for this class of equations must be the invariant measure of the corresponding limit equations.

Keywords

Invariant Borel probability measureslimit behaviorspullback attractors

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35B41: Attractors 35Q30: Navier-Stokes equations 76D03: Existence, uniqueness, and regularity theory 76D05: Navier-Stokes equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 39
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abergel, F.. Attractors for a Navier-Stokes flow in an unbounded domain. ESAIM: Math. Model. Numer. Anal. 23 (1989), 359–370.CrossRefGoogle Scholar
Abergel, F.. Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains. J. Differential Equations. 83 (1990), 85–108.CrossRefGoogle Scholar
Anh, C. T. and Trang, P. T.. Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A. 143 (2013), 223–251.CrossRefGoogle Scholar
Anh, C. T. and Trang, P. T.. On the 3D Kelvin-Voigt-Brinkman-Forchheimer equations in some unbounded domains. Nonlinear Anal. 89 (2013), 36–54.CrossRefGoogle Scholar
Babin, A. V.. The attractor of a Navier-Stokes system in an unbounded channel-like domain. J. Dynam. Differential Equations. 4 (1992), 555–584.CrossRefGoogle Scholar
Bessaih, H., Garrido-Atienza, M. J. and Schmalfuß, B.. On 3D Navier-Stokes equations: regularization and uniqueness by delays. Phys. D. 376 (2018), 228–237.CrossRefGoogle Scholar
Cao, Y., Lunasin, E. M. and Titi, E. S.. Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4 (2006), 823–848.CrossRefGoogle Scholar
Caraballo, T., Marín-Rubio, P. and Valero, J.. Autonomous and non-autonomous attractors for differential equations with delays. J. Differential Equations. 208 (2005), 9–41.CrossRefGoogle Scholar
Caraballo, T., Marín-Rubio, P. and Valero, J.. Attractors for differential equations with un-bounded delays. J. Differential Equations. 239 (2007), 311–342.CrossRefGoogle Scholar
Caraballo, T. and Real, J.. Navier-Stokes equations with delays. Proc. R. Soc. Lond. A. 457 (2001), 2441–2453.CrossRefGoogle Scholar
Caraballo, T. and Real, J.. Attractors for 2D-Navier-Stokes models with delays. J. Differential Equations. 205 (2004), 271–297.CrossRefGoogle Scholar
Chekroun, M. D. and Glatt-Holtz, N. E.. Invariant measures for dissipative dynamical systems:Abstract results and applications. Commun. Math. Phys. 316 (2012), 723–761.CrossRefGoogle Scholar
Chen, Z. and Yang, D.. Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations. J. Differential Equations. 353 (2023), 225–267.CrossRefGoogle Scholar
Constantin, P. and Foias, C.. Navier-Stokes Equations (University of Chicago Press, 1988).CrossRefGoogle Scholar
Da Prato, G. and Debussche, A.. 2D stochastic Navier-Stokes equations with a time-periodic forcing term. J. Dynam. Differential Equations. 20 (2008), 301–335.CrossRefGoogle Scholar
Foias, C., Manley, O., Rosa, R. and Temam, R.. Navier-Stokes Equations and Turbulence. In Encyclopedia of Mathematics and its Applications, Vol. 83 (Cambridge University Press, Cambridge, 2001).Google Scholar
García-Luengo, J., Marín-Rubio, P. and Real, J.. Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations. Nonlinearity. 25 (2012), .CrossRefGoogle Scholar
Garcia-Luengo, J., Marín-Rubio, P. and Planas, G.. Attractors for a double time-delayed 2D-Navier-Stokes model. Discrete Contin. Dyn. Syst. 34 (2014), 4085–4105.CrossRefGoogle Scholar
Garrido-Atienza, M. J. and Marín-Rubio, P.. Navier-Stokes equations with delays on unbounded domains. Nonlinear Anal. 64 (2006), 1100–1118.CrossRefGoogle Scholar
Hale, J. and Verduyn-Lunel, S. M.. Introduction to Functional Differential Equations (Springer-Verlag, New York, NY, 1993).CrossRefGoogle Scholar
Ju, N.. The H 1-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains. Nonlinearity. 13 (2000), 1227–1238.CrossRefGoogle Scholar
Lions, J. L.. Quelques MéThodes de RéSolution des Problémes aux Limites Non Linéaire (Dunod, Gauthier-Villars, Paris, 1969).Google Scholar
Liu, L. and Caraballo, T.. Analysis of a stochastic 2D-Navier-Stokes model with infinite delay. J. Dynam. Differential Equations. 31 (2019), 2249–2274.CrossRefGoogle Scholar
Liu, L., Caraballo, T. and Marín-Rubio, P.. Stability results for 2D Navier-Stokes equations with unbounded delay. J. Differential Equations. 11 (2018), 5685–5708.CrossRefGoogle Scholar
Liu, W.. Asymptotic behavior of solutions of time-delayed Burgers’ equation. Discrete Contin. Dyn. Syst. Ser. B. 2 (2002), 47–56.Google Scholar
Lukaszewicz, G., Real, J. and Robinson, J. C.. Invariant measures for dissipative dynamical systems and generalised Banach limits. J. Dynam. Differential Equations. 23 (2011), 225–250.CrossRefGoogle Scholar
Lukaszewicz, G. and Robinson, J. C.. Invariant measures for non-autonomous dissipative dynamical systems. Discrete Contin. Dyn. Syst. 34 (2014), 4211–4222.Google Scholar
Marín-Rubio, P., Márquez-Durán, A. and Real, J.. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete Contin. Dyn. Syst. 3 (2011), 779–796.CrossRefGoogle Scholar
Marín-Rubio, P. and Real, J.. Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains. Nonlinear Anal. 67 (2007), 2784–2799.CrossRefGoogle Scholar
Marín-Rubio, P. and Real, J.. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete Contin. Dyn. Syst. 26 (2010), 989–1006.CrossRefGoogle Scholar
Nirenberg, L.. An extended interpolation inequality. Annali Della Scuola Norm. Sup. Pisa. 20 (1966), 733–737.Google Scholar
Oskolkov, A. P.. The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 98–136.Google Scholar
Qin, Y. and Su, K.. Upper estimates on Hausdorff and fractal dimensions of global attractors for the 2D Navier-Stokes-Voight equations with a distributed delay. Asymptot. Anal. 111 (2019), 179–199.Google Scholar
Rosa, R.. The global attractor for the 2D Navier-Stokes flow on some unbounded domains. Nonlinear Anal. 32 (1998), 71–86.CrossRefGoogle Scholar
Shi, W., Cui, X., Li, X. and Yang, X. -G.. Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping. Discrete Contin. Dyn. Syst. Ser. B. 27 (2022), 5503–5534.CrossRefGoogle Scholar
Taniuchi, Y.. On the uniqueness of time-periodic solutions to the Navier-Stokes equations in unbounded domains. Math. Z. 261 (2009), 597–615.CrossRefGoogle Scholar
Temam, R.. Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications. Vol. 2 (North-Holland Publishing Co, Amsterdam-New York, 1979).Google Scholar
Temam, R.. Navier-Stokes Equations and Nonlinear Functional Analysis. In CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, PA, 1983).Google Scholar
Toi, V. M.. Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay. Evol. Equ. Control Theory. 10 (2021), 1007–1023.CrossRefGoogle Scholar
Wang, B.. Sufficient and necessary criteria for existence of pullback attractors for noncompact random dynamical systems. J. Differential Equations. 253 (2012), 1544–1583.CrossRefGoogle Scholar
Wang, B.. Uniform tail-ends estimates of the Navier-Stokes equations on unbounded channel-like domains. Proc. Amer. Math. Soc. 151 (2023), 4841–4853.CrossRefGoogle Scholar
Wang, J., Zhao, C. and Caraballo, T.. Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays. Commun. Nonlinear Sci. Numer. Simul. 91 (2020), .CrossRefGoogle Scholar
Yang, D., Caraballo, T. and Chen, Z.. The periodic and limiting behaviors of invariant measures for 3D globally modified Navier-Stokes equations. J. Dynam. Differential Equations. 36 (2024), 3863–3883.CrossRefGoogle Scholar
Yang, D., Chen, Z. and Caraballo, T.. Dynamics of a globally modified Navier-Stokes model with double delay. Z. Angew. Math. Phys. 73 (2022), 1–32.CrossRefGoogle Scholar
Yang, X. -G. and Yang, R.. Asymptotic stability of 3D Navier-Stokes equations with damping. Appl. Math. Lett. 116 (2021), .CrossRefGoogle Scholar
Zhao, C. and Yang, L.. Pullback attractor and invariant measures for the globally modified Navier-Stokes equations. Commun. Math. Sci. 15 (2017), 1565–1580.CrossRefGoogle Scholar