대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제20권4호
  • /
  • Pages.731-749
  • /
  • 2005
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

THE GLOBAL ATTRACTOR OF THE 2D G-NAVIER-STOKES EQUATIONS ON SOME UNBOUNDED DOMAINS

  • Kwean, Hyuk-Jin (Department of Mathematics Education Korea University) ;
  • Roh, Jai-Ok (Department of Mathematics Hallym University)
  • 발행 : 2005.10.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we study the two dimensional g-Navier­Stokes equations on some unbounded domain ${\Omega}\;{\subset}\;R^2$. We prove the existence of the global attractor for the two dimensional g-Navier­Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asymptotic compactness used by Rosa for the usual Navier-Stokes equations.

키워드

참고문헌

  1. F. Abergel, Attractor for a Navier-Stokes flow in an unbounded domain, Math. Model. Anal. 23 (1989), no. 3, 359-370 https://doi.org/10.1051/m2an/1989230303591
  2. F. Abergel, Existence and finite dimensionality of the global aitractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990), no. 1, 85-108 https://doi.org/10.1016/0022-0396(90)90070-6
  3. A. V. Babin, The attractor of a Navier-Stokes system in an unbounded channellike domain, J. Dynam. Differential Equations 4 (1992), no. 4, 555-584 https://doi.org/10.1007/BF01048260
  4. A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A 116A (1990), 221-243
  5. H. 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
  6. P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke fomulas and the dimension of the attmctor for the 2D Navier-Stokes equations, Comm. Pure Appl. Math. XXXVIII (1985), 1-27
  7. P. Constantin, C. Foias, C. Manley, and R. Temam, Determining modes and fractal dimension of turbulent flows, J. Fluid Mech. 150 (1988), 427-440
  8. P. Constantin, C. Foias, and R. Temam, Attractor representing turbulent flows, Mem. Amer. Math. Soc. 53 (1985). no. 314
  9. C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl, 58 (1979), 334-368
  10. J. K. Hale and G. Raugel, A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc. 329 (1992), 185-219 https://doi.org/10.2307/2154084
  11. D. Hundertmark, A. Laptev, and T. Weidl, New bounds on the Lieb-Thirring constants, Invent. Math. 140 (2000), no. 3, 693-704 https://doi.org/10.1007/s002220000077
  12. O. Ladyzhenskaya, On the dynamical system generated by the navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. lnst. Steklov.(POMI) 27 (1972), 91-114
  13. O. Ladyzhenskaya, On the dynamical system genemted by the runner-Stokes equations, English tranlation in J. of Soviet Math. 3 (1975),458-479 https://doi.org/10.1007/BF01084684
  14. O. Ladyzhenskaya, Attractor for Semigroup and Evolution Equations, Lezioni Lincei, Cam-bridge University Press, 1991
  15. I. Moise, R. Temam, and M. Ziane, Asymptotic analysis of the Navier-Stokes equations in thin domains, Topol. Methods Nonlinear Anal. 10 (1997), 249.--282
  16. S. Montgomery-Smith, Global regularity of the Navier-Stokes equations on thin three dimensional domains with periodic boundary condtitions, Electron. J. Differential Equations 11 (1999), 1-19 https://doi.org/10.1023/A:1021889401235
  17. G. Raugel and G. R. Sell, Naoier-Stokes equations on thin 3D domains. I. Global attmctors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), 503-568 https://doi.org/10.2307/2152776
  18. G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains II, Global regularity of spa-tially periodic solutions, in 'Nonlinear Partial Differential Equations and Their Applications', College de France Seminar, Longman, Harlow, XI (1994), 205-247
  19. J. Roh, g-Navier-Stokes equations, Thesis, University of Minnesota, 2001
  20. J. Roh, g-Navier-Stokes equations, Dynamics of the g-Navier-Stokes equations, J. Differential Equations 211 (2005), issue 2, 452-484 https://doi.org/10.1016/j.jde.2004.08.016
  21. R. Rosa, The global attmctor fro the 2D Navier-Stokes Flow in some unbounded domains, Nonlinear Analysis, Theory, Methods, and Applications 32 (1998), no. 1, 71-85
  22. G. R. Sell and Y. You, Dynamics of evolutionary equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002
  23. R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New-York, 1988
  24. R. Temam, Navier-Stokes Equations Theory and Numerical Analysis, Elsevier Sci-ence Pubilshers B. V. New York. 1979
  25. R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996), 499-546
  26. R. Temam and M. Ziane, Navier-Stokes equations in thin spherical domains, Contemp. Math. 209 (1997), 281-314 https://doi.org/10.1090/conm/209/02772

피인용 문헌

  1. Existence and finite time approximation of strong solutions to 2D g-Navier–Stokes equations vol.38, pp.3, 2013, https://doi.org/10.1007/s40306-013-0023-2
  2. On the Stationary Solutions to 2D g-Navier-Stokes Equations vol.42, pp.2, 2017, https://doi.org/10.1007/s40306-016-0180-1
  3. PULLBACK ATTRACTORS FOR 2D g-NAVIER-STOKES EQUATIONS WITH INFINITE DELAYS vol.31, pp.3, 2016, https://doi.org/10.4134/CKMS.c150186
  4. PULLBACK ATTRACTORS FOR STRONG SOLUTIONS OF 2D NON-AUTONOMOUS g-NAVIER-STOKES EQUATIONS vol.40, pp.4, 2015, https://doi.org/10.1007/s40306-014-0073-0
  5. Spectral Galerkin Method in Space and Time for the 2Dg-Navier-Stokes Equations vol.2013, 2013, https://doi.org/10.1155/2013/805685
  6. ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO 2D g-NAVIER-STOKES EQUATIONS vol.29, pp.4, 2014, https://doi.org/10.4134/CKMS.2014.29.4.505