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October  2021, 26(10): 5449-5463. doi: 10.3934/dcdsb.2020353

Norm inflation for the Boussinesq system

Zongyuan Li 1, and  Weinan Wang 2,

1. 

Department of Mathematics, Rutgers University, Hill Center - Busch Campus 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2. 

Department of Mathematics, University of Arizona, 617 N Santa Rita Ave, Tucson, AZ 85721, USA

Received  February 2020 Published  October 2021 Early access  December 2020

Fund Project: The second author was supported in part by NSF grant DMS-1907992

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 $.

Keywords: Boussinesq system, norm inflation, ill-posedness, negative order Besov spaces, plane waves.
Mathematics Subject Classification: Primary: 35B40; Secondary: 76D33.
Citation: Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5449-5463. doi: 10.3934/dcdsb.2020353
References:
[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Dai, J. 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.   Google Scholar

[10] C. R. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511608803.  Google Scholar
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D.-A. Geba, A. 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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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A. R. Nahmod, N. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. Wang, Regularity Problems for the Boussinesq Equations, Ph.D. Dissertation, University of Southern California, 2020.  Google Scholar

[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. Google Scholar

[24]

W. Wang, On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions, submitted for publication. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Dai, J. 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.   Google Scholar

[10] C. R. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511608803.  Google Scholar
[11]

D.-A. Geba, A. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. R. Nahmod, N. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. Wang, Regularity Problems for the Boussinesq Equations, Ph.D. Dissertation, University of Southern California, 2020.  Google Scholar

[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. Google Scholar

[24]

W. Wang, On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions, submitted for publication. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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