Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON WELL-POSEDNESS AND BLOW-UP CRITERION FOR THE 2D TROPICAL CLIMATE MODEL

  • Zhou, Mulan (School of Mathematics and Statistics Anhui Normal University)
  • Received : 2019.06.10
  • Accepted : 2019.09.05
  • Published : 2020.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider the Cauchy problem to the tropical climate model. We establish the global regularity for the 2D tropical climate model with generalized nonlocal dissipation of the barotropic mode and obtain a multi-logarithmical vorticity blow-up criterion for the 2D tropical climate model without any dissipation of the barotropic mode.

Keywords

References

  1. L. Agelas, Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations, Analysis & PDE 11 (2018), no. 4, 899-918. https://doi.org/10.2140/apde.2018.11.899
  2. H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), no. 7, 773-789. https://doi.org/10.1080/03605308008820154
  3. A. Cordoba and D. Cordoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511-528. https://doi.org/10.1007/s00220-004-1055-1
  4. M. Dabkowski, A. Kiselev, L. Silvestre, and V. Vicol, Global well-posedness of slightly supercritical active scalar equations, Analysis & PDE 7 (2014), no. 1, 43-72. https://doi.org/10.2140/apde.2014.7.43
  5. B. Dong, W. Wang, J. Wu, Z. Ye and H. Zhang, Global regularity for a class of 2D generalized tropical climate models, J. Differential Equations 266 (2019), no. 10, 6346-6382. https://doi.org/10.1016/j.jde.2018.11.007
  6. B. Dong, W. Wang, J. Wu, and H. Zhang, Global regularity results for the climate model with fractional dissipation, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 1, 211-229.
  7. 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
  8. J. Fan, S. Jiang, G. Nakamura, and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech. 13 (2011), no. 4, 557-571. https://doi.org/10.1007/s00021-010-0039-5
  9. J. Fan, H. Malaikah, S. Monaquel, G. Nakamura, and Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math. 175 (2014), no. 1, 127-131. https://doi.org/10.1007/s00605-014-0652-0
  10. 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 https://doi.org/10.4310/CMS.2004.v2.n4.a3
  11. A. Gill, Some simple solutions for heat-induced tropical circulation, Q. J. R. Meteorol. Soc. 106 (1980), 447-462. https://doi.org/10.1002/qj.49710644905
  12. N. Jacob, Pseudo differential operators and Markov processes. Vol. III, Imperial College Press, London, 2005. https://doi.org/10.1142/9781860947155
  13. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. https://doi.org/10.1002/cpa.3160410704
  14. C. E. Kenig, G. Ponce, and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), no. 2, 323-347. https://doi.org/10.2307/2939277
  15. H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000), no. 1, 191-200. https://doi.org/10.1007/s002200000267
  16. Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydro-dynamics with zero viscosity, Discrete Contin. Dyn. Syst. 25 (2009), no. 2, 575-583. https://doi.org/10.3934/dcds.2009.25.575
  17. P. G. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. https://doi.org/10.1201/9781420035674
  18. 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
  19. C. Ma, Z. Jiang, and R. Wan, Local well-posedness for the tropical climate model with fractional velocity diffusion, Kinet. Relat. Models 9 (2016), no. 3, 551-570. https://doi.org/10.3934/krm.2016006
  20. C. Ma and R. Wan, Spectral analysis and global well-posedness for a viscous tropical climate model with only a damp term, Nonlinear Anal. Real World Appl. 39 (2018), 554-567. https://doi.org/10.1016/j.nonrwa.2017.08.004
  21. A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmospheric Sci. 60 (2003), no. 15, 1809-1821. https://doi.org/10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2
  22. T. Matsuno, Quasi-geostrophic motions in the equatorial area, J. Meteorol. Soc. Jpn. 44 (1966), 25-42. https://doi.org/10.2151/jmsj1965.44.1_25
  23. 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
  24. X. Wu, Y. Yu, and Y. Tang, Global existence and asymptotic behavior for the 3D generalized Hall-MHD system, Nonlinear Anal. 151 (2017), 41-50. https://doi.org/10.1016/j.na.2016.11.010
  25. 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
  26. Y. Yu and Y. Tang, A new blow-up criterion for the 2D generalized tropical climate model, Bull. Malays. Math. Sci. Soc. 8 (2018), 1-16. https://doi.org/10.1007/s13373-016-0094-1
  27. Y. Yu, X. Wu, and Y. Tang, Global regularity of the 2D liquid crystal equations with weak velocity dissipation, Comput. Math. Appl. 74 (2017), no. 5, 920-933. https://doi.org/10.1016/j.camwa.2016.11.008
  28. Y. Yu, X. Wu, and Y. Tang, Global well-posedness for the 2D Boussinesq system with variable viscosity and damping, Math. Methods Appl. Sci. 41 (2018), no. 8, 3044-3061. https://doi.org/10.1002/mma.4799
  29. B. Yuan and J. Zhao, Global regularity of 2D almost resistive MHD equations, Nonlinear Anal. Real World Appl. 41 (2018), 53-65. https://doi.org/10.1016/j.nonrwa.2017.10.006
  30. Y. Zhou and J. Fan, A regularity criterion for the 2D MHD system with zero magnetic diffusivity, J. Math. Anal. Appl. 378 (2011), no. 1, 169-172. https://doi.org/10.1016/j.jmaa.2011.01.014
  31. M. Zhu, Global regularity for the tropical climate model with fractional diffusion on barotropic mode, Appl. Math. Lett. 81 (2018), 99-104. https://doi.org/10.1016/j.aml.2018.02.003