대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제34권3호
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  • Pages.819-839
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  • 2019
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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STABILIZATION OF 2D g-NAVIER-STOKES EQUATIONS

  • 투고 : 2018.06.13
  • 심사 : 2018.08.08
  • 발행 : 2019.07.31
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초록

We study the stabilization of 2D g-Navier-Stokes equations in bounded domains with no-slip boundary conditions. First, we stabilize an unstable stationary solution by using finite-dimensional feedback controls, where the designed feedback control scheme is based on the finite number of determining parameters such as determining Fourier modes or volume elements. Second, we stabilize the long-time behavior of solutions to 2D g-Navier-Stokes equations under action of fast oscillating-in-time external forces by showing that in this case there exists a unique time-periodic solution and every solution tends to this periodic solution as time goes to infinity.

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참고문헌

  1. 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. https://doi.org/10.4064/ap103-3-5
  2. C. T. Anh, D. T. Quyet, and D. T. Tinh, Existence and nite time approximation of strong solutions to 2D g-Navier-Stokes equations, Acta Math. Vietnam. 38 (2013), no. 3, 413-428. https://doi.org/10.1007/s40306-013-0023-2
  3. C. T. Anh and V. M. Toi, Stabilizing the long-time behavior of the Navier-Stokes-Voigt equations by fast oscillating-in-time forces, Bull. Pol. Acad. Sci. Math. 65 (2017), no. 2, 177-185. https://doi.org/10.4064/ba8094-9-2017
  4. A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction-diffusion paradigm, Evol. Equ. Control Theory 3(2014), no. 4, 579-594. https://doi.org/10.3934/eect.2014.3.579
  5. A. Azouani, E. Olson, and E. S. Titi, Continuous data assimilation using general inter-polant observables, J. Nonlinear Sci. 24 (2014), no. 2, 277-304. https://doi.org/10.1007/s00332-013-9189-y
  6. H.-O. Bae and J. Roh, Existence of solutions of the g-Navier-Stokes equations, Taiwanese J. Math. 8 (2004), no. 1, 85-102. https://doi.org/10.11650/twjm/1500558459
  7. J. Cyranka, P. B. Mucha, E. S. Titi, and P. Zgliczynski, Stabilizing the long-time behavior of the forced Navier-Stokes and damped Euler systems by large mean flow, Phys. D 369 (2018), 18-29. https://doi.org/10.1016/j.physd.2017.12.010
  8. J. Jiang and Y. Hou, The global attractor of g-Navier-Stokes equations with linear dampness on $R^2$, Appl. Math. Comput. 215 (2009), no. 3, 1068-1076. https://doi.org/10.1016/j.amc.2009.06.035
  9. J. Jiang and Y. 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. https://doi.org/10.1007/s10483-010-1304-x
  10. J. Jiang, Y. Hou, and 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. https://doi.org/10.1007/s10483-011-1402-x
  11. J. Jiang and X. Wang, Global attractor of 2D autonomous g-Navier-Stokes equations, Appl. Math. Mech. (English Ed.) 34 (2013), no. 3, 385-394. https://doi.org/10.1007/s10483-013-1678-7
  12. J. Kalantarova and T. Ozsari, Finite-parameter feedback control for stabilizing the complex Ginzburg-Landau equation, Systems Control Lett. 106 (2017), 40-46. https://doi.org/10.1016/j.sysconle.2017.06.004
  13. V. K. Kalantarov and E. S. Titi, Finite-parameters feedback control for stabilizing damped nonlinear wave equations, in Nonlinear analysis and optimization, 115-133, Contemp. Math., 659, Amer. Math. Soc., Providence, RI, 2016. https://doi.org/10.1090/conm/659/13193
  14. V. K. Kalantarov and E. S. Titi, Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), no. 3, 1325-1345. https://doi.org/10.3934/dcdsb.2018153
  15. 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. https://doi.org/10.1016/j.jmaa.2005.04.050
  16. H. Kwean, The $H^1$-compact global attractor of two-dimensional g-Navier-Stokes equations, Far East J. Dyn. Syst. 18 (2012), no. 1, 1-20.
  17. 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. https://doi.org/10.4134/CKMS.2005.20.4.731
  18. D. T. Quyet, Asymptotic behavior of strong solutions to 2D g-Navier-Stokes equations, Commun. Korean Math. Soc. 29 (2014), no. 4, 505-518. https://doi.org/10.4134/CKMS.2014.29.4.505
  19. 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. https://doi.org/10.1007/s40306-016-0180-1
  20. J. C. Robinson, Innite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
  21. J. Roh, Dynamics of the g-Navier-Stokes equations, J. Differential Equations 211 (2005), no. 2, 452-484. https://doi.org/10.1016/j.jde.2004.08.016
  22. D. Wu and J. Tao, The exponential attractors for the g-Navier-Stokes equations, J. Funct. Spaces Appl. 2012 (2012), Art. ID 503454, 12 pp.