Skip to main content
SpringerLink
Log in

A global regularity result for the 2D Boussinesq equations with critical dissipation

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by Λαu in the velocity equation and by Λβθ in the temperature equation, where \(\Lambda - \sqrt { - \Delta } \) denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when (α, β) is in the critical range: \(\alpha > (\sqrt {1777} - 23)/24 = 0.789103...\), β > 0, and α + β = 1. This result improves previous work which obtained the global regularity for \(\alpha > (23-\sqrt {145})/12 \approx 0.9132,\;\beta>0\), and α + β = 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Adhikari, C. Cao and J. Wu, The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Differential Equations 249 (2010), 1078–1088.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Adhikari, C. Cao, and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differential Equations 251 (2011), 1637–1655.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Adhikari, C. Cao, J. Wu, and X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Differential Equations 256 (2014), 3594–3613.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011.

    Book  MATH  Google Scholar 

  5. J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

    Book  MATH  Google Scholar 

  6. M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophy. J. R. Astr. Soc. 13 (1967), 529–539.

    Article  Google Scholar 

  7. C. Cao and J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal. 208 (2013), 985–1004.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006), 497–513.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Choi, T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao, On the finite-time blowup of a 1D model for the 3D axisymmetric Euler equations, Comm. Pure Appl. Math 70 (2017), 2218–2243.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Choi, A. Kiselev, and Y. Yao, Finite time blow up for a 1D model of 2D Boussinesq system, Commun. Math. Phys. 334 (2015), 1667–1679.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, Springer, Berlin, 2006, pp. 1–43.

    MATH  Google Scholar 

  12. P. Constantin and C. R. Doering, Infinite Prandtl number convection, J. Statistical Physics 94 (1999), 159–172.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Constantin, A. Majda, and E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity 7 (1994), 1495–1533.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal. 22 (2012), 1289–1321.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Córdoba and D. Córdoba, A maximum princple applied to quasi-geostroohhic equations, Comm. Math. Phys. 249 (2004), 511–528.

    Article  MathSciNet  MATH  Google Scholar 

  16. X. Cui, C. Dou, and Q. Jiu, Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces, J. Partial Differential Equations 25 (2012), 220–238.

    MathSciNet  MATH  Google Scholar 

  17. R. Danchin, Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proc. Amer. Math. Soc. 141 (2013), 1979–1993.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Danchin and M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci. 21 (2011), 421–457.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Doering and J. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  20. A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982.

    Google Scholar 

  21. L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations 39 (2014), 1128–1157.

    Article  MathSciNet  MATH  Google Scholar 

  22. I. Held, R. Pierrehumbert, S. Garner, and K. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech. 282 (1995), 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Hittmeir and S. Merino-Aceituno, Kinetic derivation of fractional Stokes and Stokes-Fourier systems, Kinet. and Relat Models 9 (2016), 105–129.

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Hmidi, On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal Partial Differential Equations 4 (2011), 247–284.

    MathSciNet  MATH  Google Scholar 

  25. T. Hmidi, S. Keraani, and F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations 249 (2010), 2147–2174.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. Hmidi, S. Keraani, and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations 36 (2011), 420–445.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete and Cont. Dyn. Syst. 12 (2005), 1–12.

    MathSciNet  MATH  Google Scholar 

  28. J. Jia, J. Peng, and K. Li, On the global well-posedness of a generalized 2D Boussinesq equations, NoDEA Nonlinear Differential Equations Appl. 2 (2015), 911–945.

    Article  MathSciNet  MATH  Google Scholar 

  29. Q. Jiu, C. Miao, J. Wu, and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal. 46 (2014), 3426–3454.

    Article  MathSciNet  MATH  Google Scholar 

  30. Q. Jiu, J. Wu, and W. Yang, Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation, J. Nonlinear Science 25 (2014), 37–58.

    Article  MathSciNet  MATH  Google Scholar 

  31. D.KC, D. Regmi, L. Tao, and J. Wu, The 2D Euler-Boussinesq equations with a singular velocity, J. Differential Equations 257 (2014), 82–108.

    Article  MathSciNet  MATH  Google Scholar 

  32. C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de-Vries equation via the contraction principle, Comm. Pure App. Math. 46 (1993), 527–620.

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Lai, R. Pan, and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal. 199 (2011), 739–760.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Larios, E. Lunasin, and E.S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations 255 (2013), 2636–2654.

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Luo and T. Hou, Potentially singular solutions of the 3D incompressible Euler equations, Proc. Natl. Acad. Sci. USA 111 (2014), 12968–12973.

    Article  Google Scholar 

  36. A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics 9, AMS/CIMS, 2003.

    Book  MATH  Google Scholar 

  37. A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001.

    Book  MATH  Google Scholar 

  38. C. Miao and L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl. 18 (2011), 707–735.

    Article  MathSciNet  MATH  Google Scholar 

  39. C. Miao, J. Wu, and Z. Zhang, Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics, Science Press, Beijing, China, 2012 (in Chinese).

    Google Scholar 

  40. L. Nirengerg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959), 115–162.

    MathSciNet  Google Scholar 

  41. K. Ohkitani, Comparison between the Boussinesq and coupled Euler equations in two dimensions. Tosio Kato’s method and principle for evolution equations in mathematical physics, (Sapporo, 2001) Surikaisekikenkyusho Kokyuroku No. 1234 (2001), 127–145.

    Google Scholar 

  42. J. Pedlosky, Geophysical Fluid Dyanmics, Springer-Verlag, New York, 1987.

    Book  MATH  Google Scholar 

  43. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996.

    MATH  Google Scholar 

  44. A. Sarria and J. Wu, Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations, J. Differential Equations 259 (2015), 3559–3576.

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

  46. H. Triebel, Theory of Function Spaces II, Birkhäuser Verlag, 1992.

    Book  MATH  Google Scholar 

  47. B. Wen, N. Dianati, E. Lunasin, G. Chini, and C. Doering, New upper bounds and reduced dynamical modeling for Rayleigh-Bénard convection in a fluid saturated porous layer, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 2191–2199.

    Article  MathSciNet  Google Scholar 

  48. J. Whitehead and C. Doering, Internal heating driven convection at infinite Prandtl number, J. Math. Phys. 52 (2011), 093101, 11 pp.

    Article  MathSciNet  MATH  Google Scholar 

  49. J. Wu and X. Xu, Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation, Nonlinearity 27 (2014), 2215–2232.

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Wu, X. Xu, and Z. YeThe 2D Boussinesq equations with partial or fractional dissipation, J. Pures Appl. Math. 115 (2018), 187–217.

    Article  Google Scholar 

  51. J. Wu, X. Xu and Z. Ye, Global smooth solutions to the n-dimensional damped models of incompressible fluid mechanics with small initial datum, J. Nonlinear Science 5 (2015), 157–192.

    Article  MathSciNet  MATH  Google Scholar 

  52. X. Xu, Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Analysis: TMA 72 (2010), 677–681.

    Article  MathSciNet  MATH  Google Scholar 

  53. W. Yang, Q. Jiu, and J. Wu, Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, J. Differential Equations 257 (2014), 4188–4213.

    Article  MathSciNet  MATH  Google Scholar 

  54. Z. Ye and X. Xu, Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system, Nonlinear Anal. 100 (2014), 86–96.

    Article  MathSciNet  MATH  Google Scholar 

  55. K. Zhao, 2D inviscid heat conductive Boussinesq equations on a bounded domain, Michigan Math. J. 59 (2010), 329–352.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045-7594, USA

    Atanas Stefanov

  2. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Jiahong Wu

Authors
  1. Atanas Stefanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiahong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiahong Wu.

Additional information

Partially supported by NSF grant DMS 1313107.

Partially supported by NSF grant DMS 1209153 and the AT&T Foundation at Oklahoma State University.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stefanov, A., Wu, J. A global regularity result for the 2D Boussinesq equations with critical dissipation. JAMA 137, 269–290 (2019). https://doi.org/10.1007/s11854-018-0073-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-018-0073-4

Search

Navigation