Startseite On the stability of solutions to stochastic 2D g-Navier–Stokes equations with finite delays
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

On the stability of solutions to stochastic 2D g-Navier–Stokes equations with finite delays

  • Cung The Anh EMAIL logo , Nguyen Van Thanh und Nguyen Viet Tuan
Veröffentlicht/Copyright: 27. Oktober 2017
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

In this paper, we study the exponential mean square stability and almost sure exponential stability of weak solutions to the stochastic 2D g-Navier–Stokes equations with finite delays in bounded domains.

MSC 2010: 35B35; 35Q35; 35D35

Communicated by Vyacheslav L. Girko


Award Identifier / Grant number: B2016-SPH-17

Funding statement: This work was supported by the Vietnam Ministry of Education and Training under grant number B2016-SPH-17.

References

[1] C. T. Anh and N. T. Da, The exponential behaviour and stabilizability of stochastic 2D hydrodynamical type systems, Stochastics 89 (2017), no. 3–4, 593–618. 10.1080/17442508.2016.1269767Suche in Google Scholar

[2] C. T. Anh and D. T. Quyet, g-Navier–Stokes equations with infinite delays, Vietnam J. Math. 40 (2012), no. 1, 57–78. Suche in Google Scholar

[3] C. T. Anh and D. T. Quyet, Long-time behavior for 2D non-autonomous g-Navier–Stokes equations, Ann. Polon. Math. 103 (2012), no. 3, 277–302. 10.4064/ap103-3-5Suche in Google Scholar

[4] H.-O. Bae and J. Roh, Existence of solutions of the g-Navier–Stokes equations, Taiwanese J. Math. 8 (2004), no. 1, 85–102. 10.11650/twjm/1500558459Suche in Google Scholar

[5] H. Breckner, Approximation of the solution of the stochastic Navier–Stokes equation, Optimization 49 (2001), no. 1–2, 15–38. 10.1080/02331930108844518Suche in Google Scholar

[6] T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier–Stokes models with delays, Dyn. Partial Differ. Equ. 11 (2014), no. 4, 345–359. 10.4310/DPDE.2014.v11.n4.a3Suche in Google Scholar

[7] T. Caraballo and X. Han, A survey on Navier–Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S 8 (2015), no. 6, 1079–1101. 10.3934/dcdss.2015.8.1079Suche in Google Scholar

[8] T. Caraballo, J. A. Langa and T. Taniguchi, The exponential behaviour and stabilizability of stochastic 2D-Navier–Stokes equations, J. Differential Equations 179 (2002), no. 2, 714–737. 10.1006/jdeq.2001.4037Suche in Google Scholar

[9] T. Caraballo and J. Real, Navier–Stokes equations with delays, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2014, 2441–2453. 10.1098/rspa.2001.0807Suche in Google Scholar

[10] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier–Stokes equations with delays, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2040, 3181–3194. 10.1098/rspa.2003.1166Suche in Google Scholar

[11] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier–Stokes equations with delays, Proc. Indian Acad. Sci. Math. Sci. 122 (2012), no. 2, 283–295. 10.1007/s12044-012-0071-xSuche in Google Scholar

[12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, 1992. 10.1017/CBO9780511666223Suche in Google Scholar

[13] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights, Elsevier, Amsterdam, 2014. 10.1016/B978-0-12-800882-9.00004-4Suche in Google Scholar

[14] M. J. Garrido-Atienza and P. Marín-Rubio, Navier–Stokes equations with delays on unbounded domains, Nonlinear Anal. 64 (2006), no. 5, 1100–1118. 10.1016/j.na.2005.05.057Suche in Google Scholar

[15] J. Jiang and Y. Hou, The global attractor of g-Navier–Stokes equations with linear dampness on 𝐑2, Appl. Math. Comput. 215 (2009), no. 3, 1068–1076. 10.1016/j.amc.2009.06.035Suche in Google Scholar

[16] J.-P. Jiang and Y.-R. Hou, Pullback attractor of 2D non-autonomous g-Navier–Stokes equations on some bounded domains, Appl. Math. Mech. (English Ed.) 31 (2010), no. 6, 697–708. 10.1007/s10483-010-1304-xSuche in Google Scholar

[17] J.-P. Jiang, Y.-R. Hou and X.-X. Wang, Pullback attractor of 2D nonautonomous g-Navier–Stokes equations with linear dampness, Appl. Math. Mech. (English Ed.) 32 (2011), no. 2, 151–166. 10.1007/s10483-011-1402-xSuche in Google Scholar

[18] J.-P. Jiang and X.-X. Wang, Global attractor of 2D autonomous g-Navier–Stokes equations, Appl. Math. Mech. (English Ed.) 34 (2013), no. 3, 385–394. 10.1007/s10483-013-1678-7Suche in Google Scholar

[19] M. Kwak, H. Kwean and J. Roh, The dimension of attractor of the 2D g-Navier–Stokes equations, J. Math. Anal. Appl. 315 (2006), no. 2, 436–461. 10.1016/j.jmaa.2005.04.050Suche in Google Scholar

[20] H. Kwean, The H1-compact global attractor of two-dimensional g-Navier–Stokes equations, Far East J. Dyn. Syst. 18 (2012), no. 1, 1–20. Suche in Google Scholar

[21] H. Kwean and J. Roh, The global attractor of the 2D g-Navier–Stokes equations on some unbounded domains, Commun. Korean Math. Soc. 20 (2005), no. 4, 731–749. 10.4134/CKMS.2005.20.4.731Suche in Google Scholar

[22] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier–Stokes model in an infinite delay case, Nonlinear Anal. 74 (2011), no. 5, 2012–2030. 10.1016/j.na.2010.11.008Suche in Google Scholar

[23] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier–Stokes equations, Discrete Contin. Dyn. Syst. 21 (2008), no. 4, 1245–1258. 10.3934/dcds.2008.21.1245Suche in Google Scholar

[24] D. T. Quyet, Asymptotic behavior of strong solutions to 2D g-Navier–Stokes equations, Commun. Korean Math. Soc. 29 (2014), no. 4, 505–518. 10.4134/CKMS.2014.29.4.505Suche in Google Scholar

[25] D. T. Quyet, Pullback attractors for strong solutions of 2D non-autonomous g-Navier–Stokes equations, Acta Math. Vietnam. 40 (2015), no. 4, 637–651. 10.1007/s40306-014-0073-0Suche in Google Scholar

[26] D. T. Quyet, Pullback attractors for 2D g-Navier–Stokes equations with infinite delays, Commun. Korean Math. Soc. 31 (2016), no. 3, 519–532. 10.4134/CKMS.c150186Suche in Google Scholar

[27] D. T. Quyet and N. V. Tuan, On the stationary solutions to 2D g-Navier–Stokes equations, Acta Math. Vietnam. 42 (2017), no. 2, 357–367. 10.1007/s40306-016-0180-1Suche in Google Scholar

[28] J. Roh, g-Navier–Stokes equations, Ph.D. Thesis, University of Minnesota, 2001. Suche in Google Scholar

[29] J. Roh, Dynamics of the g-Navier–Stokes equations, J. Differential Equations 211 (2005), no. 2, 452–484. 10.1016/j.jde.2004.08.016Suche in Google Scholar

[30] S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (2006), no. 11, 1636–1659. 10.1016/j.spa.2006.04.001Suche in Google Scholar

[31] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst. 12 (2005), no. 5, 997–1018. 10.3934/dcds.2005.12.997Suche in Google Scholar

[32] R. Temam, Navier–Stokes Equations, Stud. Math. Appl. 2, North-Holland Publishing Co., Amsterdam, 1979. Suche in Google Scholar

[33] L. Wan and Q. Zhou, Asymptotic behaviors of stochastic two-dimensional Navier–Stokes equations with finite memory, J. Math. Phys. 52 (2011), no. 4, Article ID 042703. 10.1063/1.3574630Suche in Google Scholar

[34] M. J. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier–Stokes equations with time delay, Appl. Math. J. Chinese Univ. Ser. A 24 (2009), no. 4, 493–500. Suche in Google Scholar

[35] D. Wu, The finite-dimensional uniform attractors for the nonautonomous g-Navier–Stokes equations, J. Appl. Math. 2009 (2009), Article ID 150420. 10.1155/2009/150420Suche in Google Scholar

[36] D. Wu and J. Tao, The exponential attractors for the g-Navier–Stokes equations, J. Funct. Spaces Appl. 2012 (2012), Article ID 503454. 10.1155/2012/503454Suche in Google Scholar

Received: 2017-5-5
Accepted: 2017-8-12
Published Online: 2017-10-27
Published in Print: 2017-12-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 5.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/rose-2017-0016/html
Button zum nach oben scrollen