Norm inflation for the Boussinesq system

Zongyuan Li; Weinan Wang

doi:10.3934/dcdsb.2020353

Article Contents

2021, Volume 26, Issue 10: 5449-5463. Doi: 10.3934/dcdsb.2020353

This issue Previous Article Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise Next Article On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type

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: January 31, 2020

Early access: November 2020

Published: October 2021

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

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Abstract

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 $.

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

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[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. 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.
[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.
[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.
[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. 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.
[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.