REVIEW ARTICLE
Qualitative properties for the 2- nonautonomous stochastic Navier–Stokes equations
Dingshi Li,
Shaoyue Mi,
Dingshi Li
School of Mathematics, Southwest Jiaotong University, Chengdu, P. R. China
Search for more papers by this authorCorresponding Author
Shaoyue Mi
School of Mathematics, Southwest Jiaotong University, Chengdu, P. R. China
Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain
Correspondence
Shaoyue Mi, School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, P. R. China.
Email: [email protected]
Search for more papers by this authorDingshi Li,
Shaoyue Mi,
Dingshi Li
School of Mathematics, Southwest Jiaotong University, Chengdu, P. R. China
Search for more papers by this authorCorresponding Author
Shaoyue Mi
School of Mathematics, Southwest Jiaotong University, Chengdu, P. R. China
Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain
Correspondence
Shaoyue Mi, School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, P. R. China.
Email: [email protected]
Search for more papers by this authorFirst published: 25 June 2025
Abstract
We establish the pullback asymptotic compact of the family probability measures with respect to probability distributions of the solutions of the 2- nonautonomous stochastic Navier–Stokes equations, and prove the existence and uniqueness of a pullback measure attractor. The structures of pullback measure attractors is characterized by complete solutions, which is an extension of the notation of evolution systems of measures introduced and developed by Da Prato and Röckner in [Rendiconti Lincei-Matematica e Applicazioni 17 (2006), no. 4, 397–403] and [Seminar on Stochastic Analysis, Random Fields and Applications, 115–122, Springer, 2007]. Moreover, for stochastic systems containing periodic deterministic forcing terms, we show the pullback measure attractors are also periodic under certain conditions.
CONFLICT OF INTEREST STATEMENT
All authors declare no competing interests.
REFERENCES
- 1Z. Brzeźniak, T. Caraballo, J. Langa, Y. Li, G. Łukaszewicz, and J. Real, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differential Equations 255 (2013), 3897–3919.
- 2T. Caraballo, G. Łukaszewicz, and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal. Theory Methods Appl. 64 (2006), 484–498.
- 3T. Caraballo, G. Łukaszewicz, and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris 342 (2006), 263–268.
10.1016/j.crma.2005.12.015 Google Scholar
- 4V. Chepyzhov and M. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor, J. Dynam. Differential Equations 19 (2007), 655–684.
- 5I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal. 188 (2008), 117–153.
- 6I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its -approximation, Phys. D 237 (2008), 1352–1367.
10.1016/j.physd.2008.03.012 Google Scholar
- 7I. Chueshov and A. Millet, Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem, Stoch Anal Appl. 29 (2011), 570–611.
- 8H. Crauel, A. Debussche, and F. Flandoli, Random attractors, J. Dynam. Differential Equations 9 (1997), 307–341.
10.1007/BF02219225 Google Scholar
- 9H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), 365–393.
- 10H. Cui, M. M. Freitas, and J. A. Langa, Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), 1297.
- 11G. Da Prato and A. Debussche, 2D stochastic Navier-Stokes equations with a time-periodic forcing term, J. Dynam. Differential Equations 20 (2008), 301–335.
- 12G. Da Prato and M. Röckner, Dissipative stochastic equations in Hilbert space with time dependent coefficients, Rendiconti Lincei-Matematica e Applicazioni 17 (2006), 397–403.
- 13G. Da Prato and M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, Seminar on Stochastic Analysis, Random Fields and Applications V, Springer, Biblioteca de la Universidad de Sevilla, 2007, 115–122.
10.1007/978-3-7643-8458-6_7 Google Scholar
- 14F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl. 1 (1994), 403–423.
10.1007/BF01194988 Google Scholar
- 15F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields 102 (1995), 367–391.
- 16F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Commun. Math. Phys. 172 (1995), 119–141.
- 17F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics 59 (1996), 21–45.
- 18B. Gess, W. Liu, and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations 269 (2020), 3414–3455.
- 19N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations 14 (2009), no. 3, 567–600.
- 20M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math. (2006), 164, no. 3, 993–1032.
- 21Y. Hou and K. Li, The uniform attractor for the 2D non-autonomous Navier-Stokes flow in some unbounded domain, Nonlinear Anal. Theory Methods Appl. 58 (2004), 609–630.
- 22O. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, J. Sov. Math. 3 (1975), 458–479.
10.1007/BF01084684 Google Scholar
- 23J. A. Langa, G. Łukaszewicz, and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains, Nonlinear Anal. Theory Methods Appl. 66 (2007), 735–749.
- 24D. Li, Y. Lin, and Z. Pu, Non-autonomous stochastic lattice systems with Markovian switching, Discrete Contin. Dyn. Syst. 43 (2023), 1860–1877.
- 25D. Li and B. Wang, Pullback measure attractors for non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations 397 (2024), 232–261.
- 26D. Li, B. Wang, and X. Wang, Periodic measures of stochastic delay lattice systems, J. Differential Equations 272 (2021), 74–104.
- 27R. Liu and K. Lu, Statistical properties of 2D stochastic Navier-Stokes equations with time-periodic forcing and degenerate stochastic forcing, arXiv preprint arXiv:2105.00598 (2021), https://doi.org/10.48550/arXiv.2105.00598.
10.48550/arXiv.2105.00598 Google Scholar
- 28C. Marek and N. J. Cutland, Measure attractors for stochastic Navier-Stokes equations, Electron. J. Probab. 8 (1998), 1–15.
- 29J. C. Mattingly, Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity, Commun. Math. Phys. 206 (1999), 273–288.
- 30J. C. Mattingly, Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics, Commun. Math. Phys. 230 (2002), 421–462.
- 31M. Röckner and X. Zhang, Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity, Probab. Theory Related Fields 145 (2009), 211–267.
- 32B. Schmalfuß, Long-time behaviour of the stochastic Navier-Stokes equation, Math. Nachr. 152 (1991), 7–20.
- 33B. Schmalfuß, Measure attractors of the stochastic Navier-Stokes equation, Universität Bremen. Fachbereiche Mathematik/Informatik, Elektrotechnik, 1991.
- 34B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, Dresden, 1992, 185–192.
- 35B. Schmalfuß, Qualitative properties for the stochastic Navier-Stokes equation, Nonlinear Anal. Theory Methods Appl. 28 (1997), 1545–1563.
- 36B. Schmalfuß, Attractors for the non–autonomous dynamical systems, in Equadiff 99 (2 Volumes), World Scientific, World Scientific, 2000, 684–689.
- 37S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (2006), 1636–1659.
- 38R. Temam, Navier-Stokes equations: theory and numerical analysis (book), Studies in Mathematics and Its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1977.
- 39B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electron. J. Differential Equations 59 (2012), 18.
- 40B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations 253 (2012), 1544–1583.
- 41B. Wang, Uniform tail-ends estimates of the Navier-Stokes equations on unbounded channel-like domains, Proc. Amer. Math. Soc. 151 (2023), 4841–4853.
- 42R. Wang, T. Caraballo, and N. H. Tuan, Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: theoretical results and applications, Proc. Amer. Math. Soc. 151 (2023), 2449–2458.
