Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE H1-UNIFORM ATTRACTOR FOR THE 2D NON-AUTONOMOUS TROPICAL CLIMATE MODEL ON SOME UNBOUNDED DOMAINS

  • Pigong, Han (Academy of Mathematics and Systems Science Chinese Academy of Sciences) ;
  • Keke, Lei (Academy of Mathematics and Systems Science Chinese Academy of Sciences) ;
  • Chenggang, Liu (School of Statistics and Mathematics Zhongnan University of Economics and Law) ;
  • Xuewen, Wang (Academy of Mathematics and Systems Science Chinese Academy of Sciences)
  • Received : 2021.11.03
  • Accepted : 2022.04.22
  • Published : 2022.11.30
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Abstract

In this paper, we study the uniform attractor of the 2D nonautonomous tropical climate model in an arbitrary unbounded domain on which the Poincaré inequality holds. We prove that the uniform attractor is compact not only in the L2-spaces but also in the H1-spaces. Our proof is based on the concept of asymptotical compactness. Finally, for the quasiperiodical external force case, the dimension estimates of such a uniform attractor are also obtained.

Keywords

Acknowledgement

This work is supported by the National Key R&D Program of China (2021YFA1000800), the National Natural Science Foundation of China under Grant No. 11871457, the K. C. Wong Education Foundation, Chinese Academy of Sciences.

References

  1. F. Abergel, Attractor for a Navier-Stokes flow in an unbounded domain, RAIRO Model. Math. Anal. Numer. 23 (1989), no. 3, 359-370. https://doi.org/10.1051/m2an/1989230303591
  2. C. Ai, Z. Tan, and J. Zhou, Global well-posedness and existence of uniform attractor for magnetohydrodynamic equations, Math. Methods Appl. Sci. 43 (2020), no. 12, 7045-7069. https://doi.org/10.1002/mma.6414
  3. A. V. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynam. Differential Equations 4 (1992), no. 4, 555-584. https://doi.org/10.1007/BF01048260
  4. H.-O. Bae and B. J. Jin, Temporal and spatial decays for the Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 3, 461-477. https://doi.org/10.1017/S0308210500003966
  5. H.-O. Bae and B. J. Jin, Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations, J. Differential Equations 209 (2005), no. 2, 365-391. https://doi.org/10.1016/j.jde.2004.09.011
  6. H.-O. Bae and B. J. Jin, Asymptotic behavior for the Navier-Stokes equations in 2D exterior domains, J. Funct. Anal. 240 (2006), no. 2, 508-529. https://doi.org/10.1016/j.jfa.2006.04.029
  7. H.-O. Bae and B. J. Jin, Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, Bull. Korean Math. Soc. 44 (2007), no. 3, 547-567. https://doi.org/10.4134/BKMS.2007.44.3.547
  8. H.-O. Bae and B. J. Jin, Existence of strong mild solution of the Navier-Stokes equations in the half space with nondecaying initial data, J. Korean Math. Soc. 49 (2012), no. 1, 113-138. https://doi.org/10.4134/JKMS.2012.49.1.113
  9. V. V. Chepyzhov and M. A. Efendiev, Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: an example, Comm. Pure Appl. Math. 53 (2000), no. 5, 647-665. https://doi.org/10.1002/(SICI)1097-0312(200005)53:5<647::AID-CPA5>3.0.CO;2-#
  10. V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. http://dx.doi.org/10.1090/coll/049
  11. P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Comm. Pure Appl. Math. 38 (1985), no. 1, 1-27. https://doi.org/10.1002/cpa.3160380102
  12. B.-Q. Dong, J. Wu, and Z. Ye, Global regularity for a 2D tropical climate model with fractional dissipation, J. Nonlinear Sci. 29 (2019), no. 2, 511-550. https://doi.org/10.1007/s00332-018-9495-5
  13. L. C. Evans, Partial differential equations, second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. https://doi.org/10.1090/gsm/019
  14. D. M. W. Frierson, A. J. Majda, and O. M. Pauluis, Large scale dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit, Commun. Math. Sci. 2 (2004), no. 4, 591-626. http://projecteuclid.org/euclid.cms/1109885499 109885499
  15. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, second edition, Springer Monographs in Mathematics, Springer, New York, 2011. https://doi.org/10.1007/978-0-387-09620-9
  16. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
  17. D. Gong, H. Song, and C. Zhong, Attractors for nonautonomous two-dimensional space periodic Navier-Stokes equations, J. Math. Phys. 50 (2009), no. 10, 102706, 10 pp. https://doi.org/10.1063/1.3227652
  18. A. Haraux, Attractors of asymptotically compact processes and applications to nonlinear partial differential equations, Comm. Partial Differential Equations 13 (1988), no. 11, 1383-1414. https://doi.org/10.1080/03605308808820580
  19. J. G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math. 136 (1976), no. 1-2, 61-102. https://doi.org/10.1007/BF02392043
  20. J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), no. 5, 639-681. https://doi.org/10.1512/iumj.1980.29.29048
  21. Y. Hou and K. Li, The uniform attractor for the 2D non-autonomous Navier-Stokes flow in some unbounded domain, Nonlinear Anal. 58 (2004), no. 5-6, 609-630. https://doi.org/10.1016/j.na.2004.02.031
  22. N. Ju, The H1-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains, Nonlinearity 13 (2000), no. 4, 1227-1238. https://doi.org/10.1088/0951-7715/13/4/313
  23. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, second English edition, revised and enlarged., translated from the Russian by Richard A. Silverman and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York, 1969.
  24. O. A. Ladyzhenskaya, A dynamical system that is generated by the Navier-Stokes equations, Dokl. Akad. Nauk SSSR 205 (1972), 318-320.
  25. J. Li and E. Titi, Global well-posedness of strong solutions to a tropical climate model, Discrete Contin. Dyn. Syst. 36 (2016), no. 8, 4495-4516. https://doi.org/10.3934/dcds.2016.36.4495
  26. H. Li and Y. Xiao, Decay rate of unique global solution for a class of 2D tropical climate model, Math. Methods Appl. Sci. 42 (2019), no. 8, 2533-2543. https://doi.org/10.1002/mma.5529
  27. E. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of Schrodinger equations and their relations to Sobolev inequalities, Studies in Mathematical Physics, Essays in Honour of Valentine Bargmann, 269-303, Princeton Univ. Press, Princeton, NJ. 1976.
  28. S. Lu, H. Wu, and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 701-719. https://doi.org/10.3934/dcds.2005.13.701
  29. I. Moise, R. Rosa, and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 473-496. https://doi.org/10.3934/dcds.2004.10.473
  30. R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1998), no. 1, 71-85. https://doi.org/10.1016/S0362-546X(97)00453-7
  31. V. A. Solonnikov and V. E. Scadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 125 (1973), 196-210, 235.
  32. R. Temam, Navier-Stokes equations and nonlinear functional analysis, second edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. https://doi.org/10. 1137/1.9781611970050 https://doi.org/10.1137/1.9781611970050
  33. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4612-0645-3
  34. R. Wan, Global small solutions to a tropical climate model without thermal diffusion, J. Math. Phys. 57 (2016), no. 2, 021507, 13 pp. https://doi.org/10.1063/1.4941039
  35. H. Xie and Z. Zhang, Time decay rate of solutions to the tropical climate model equations in Rn, Appl. Anal. 100 (2021), no. 7, 1487-1500. https://doi.org/10.1080/00036811.2019.1646422
  36. Z. Ye, Global regularity for a class of 2D tropical climate model, J. Math. Anal. Appl. 446 (2017), no. 1, 307-321. https://doi.org/10.1016/j.jmaa.2016.08.053