\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Norm inflation for the Boussinesq system

The second author was supported in part by NSF grant DMS-1907992
Abstract Full Text(HTML) Related Papers Cited by
  • We prove the norm inflation phenomena for the Boussinesq system on $ \mathbb T^3 $. For arbitrarily small initial data $ (u_0,\rho_0) $ in the negative-order Besov spaces $ \dot{B}^{-1}_{\infty, \infty} \times \dot{B}^{-1}_{\infty, \infty} $, the solution can become arbitrarily large in a short time. Such largeness can be detected in $ \rho $ in Besov spaces of any negative order: $ \dot{B}^{-s}_{\infty, \infty} $ for any $ s>0 $.

    Mathematics Subject Classification: Primary: 35B40; Secondary: 76D33.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.  doi: 10.1016/j.jfa.2008.07.008.
    [2] L. Brandolese and J. He, Uniqueness theorems for the Boussinesq system, Tohoku Math. J. (2), 72 (2020), 283-297.  doi: 10.2748/tmj/1593136822.
    [3] L. Brandolese and C. Mouzouni, A short proof of the large time energy growth for the Boussinesq system, J. Nonlinear Sci., 27 (2017), 1589-1608.  doi: 10.1007/s00332-017-9379-0.
    [4] C. Cao and J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.
    [5] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.
    [6] R. M. Chen and Y. Liu, On the ill-posedness of a weakly dispersive one-dimensional Boussinesq system, J. Anal. Math., 121 (2013), 299-316.  doi: 10.1007/s11854-013-0037-7.
    [7] A. Cheskidov and M. Dai, Norm inflation for generalized Navier-Stokes equations, Indiana Univ. Math. J., 63 (2014), 869-884.  doi: 10.1512/iumj.2014.63.5249.
    [8] A. Cheskidov and M. Dai, Norm inflation for generalized Magneto-hydrodynamic system, Nonlinearity, 28 (2015), 129-142.  doi: 10.1088/0951-7715/28/1/129.
    [9] M. DaiJ. Qing and M. E. Schonbek, Norm inflation for incompressible magneto-hydrodynamic system in $\dot{B}^{-1}_{\infty, \infty}$  , Adv. Differential Equations, 16 (2011), 725-746. 
    [10] C. R. Doering and  J. D. GibbonApplied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511608803.
    [11] D.-A. GebaA. A. Himonasb and D. Karapetyana, Ill-posedness results for generalized Boussinesq equations, Non. Anal., 95 (2014), 404-413.  doi: 10.1016/j.na.2013.09.017.
    [12] T. Hmidi and S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461-480. 
    [13] T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.
    [14] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.
    [15] I. Kukavica and W. Wang, Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity, Pure Appl. Funct. Anal., 5 (2020), 27-45. 
    [16] I. Kukavica and W. Wang, Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity, J. Dynam. Differential Equations, 32 (2020), 2061-2077.  doi: 10.1007/s10884-019-09802-w.
    [17] P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.
    [18] A. R. NahmodN. Pavlović and G. Staffilani, Almost sure existence of global weak solutions for supercritical Navier-Stokes equation, SIAM J. Math. Anal., 45 (2013), 3431-3452.  doi: 10.1137/120882184.
    [19] A. Stefanov and J. Wu, A global regularity result for the 2D Boussinesq equations with critical dissipation, J. Anal. Math., 137 (2019), 269-290.  doi: 10.1007/s11854-018-0073-4.
    [20] R. Temam, Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, reprint of the 1984 edition. doi: 10.1090/chel/343.
    [21] W. Wang, On the global regularity for a 3D Boussinesq model without thermal diffusion, Z. Angew. Math. Phys., 70 (2019), Paper No. 174, 6 pp. doi: 10.1007/s00033-019-1221-0.
    [22] W. Wang, Regularity Problems for the Boussinesq Equations, Ph.D. Dissertation, University of Southern California, 2020.
    [23] W. Wang, On the analyticity and Gevrey regularity of solutions to the three-dimensional inviscid Boussinesq equations in a half space, submitted for publication.
    [24] W. Wang, On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions, submitted for publication.
    [25] W. Wang and H. Yue, Time decay of almost-sure global weak solutions to the Navier–Stokes and the MHD equations with initial data in negative-order Sobolev spaces, submitted for publication.
    [26] W. Wang and H. Yue, Almost sure existence of global weak solutions for the Boussinesq equations, Dyn. Partial Differ. Equ., 17 (2020), 165-183.  doi: 10.4310/DPDE.2020.v17.n2.a4.
    [27] T. Yoneda, Ill-posedness of the 3D Navier–Stokes equations in a generalized Besov space near $\rm BMO^{-1}$, J. Func. Anal., 258 (2010), 3376-3387.  doi: 10.1016/j.jfa.2010.02.005.
  • 加载中
SHARE

Article Metrics

HTML views(339) PDF downloads(220) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return