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May  2019, 39(5): 2709-2730. doi: 10.3934/dcds.2019113

Global well-posedness for the 2D Boussinesq equations with a velocity damping term

Renhui Wan

Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received  May 2018 Revised  September 2018 Published  January 2019

In this paper, we prove global well-posedness of smooth solutions to the two-dimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state $ (0, x_2) $. As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [1] (see P.3597).

Keywords: Boussinesq equations, global well-posedness, velocity damping.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76B03.
Citation: Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113
References:
[1]

D. Adhikar, C. Cao, J. Wu and X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Differential Equations, 256 (2014), 3594-3613.  doi: 10.1016/j.jde.2014.02.012.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

P. Constantin and C. R. Doering, Infinite Prandtl number convection, J. Stat. Phys., 94 (1999), 159-172.  doi: 10.1023/A:1004511312885.

[4]

T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-gepstrophic and inviscid boussinesq systems, SIAM. J. Math. Anal., 47 (2015), 4672-4684.  doi: 10.1137/14099036X.

[5] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982. 
[6]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier- Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[7]

C. 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), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[8]

A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lect. Notes Math., vol. 9, AMS/CIMS, 2003. doi: 10.1090/cln/009.

[9]

J. Pedlosky, Geoph Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

[10]

A. Pekalski and K. Sznajd-Weron (Eds.), Anomalous Diffusion, Form Basics to Applications, Lecture Notes in Phys., vol. 519, Springer-Verlag, Berlin, 1999.

[11]

X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020.

[12]

R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys., 67 (2016), Art. 104, 22 pp. doi: 10.1007/s00033-016-0697-0.

[13]

R. Wan, Global well-posedness of strong solutions to the 2D damped Boussinesq and MHD equations with large velocity, Comm. Math. Sci., 15 (2017), 1617-1626.  doi: 10.4310/CMS.2017.v15.n6.a6.

[14]

R. Wan, Long time stability for the dispersive SQG equation and Bousinessq equations in Sobolev space $H^s$, Commun. Contemp. Math., 2018. doi: 10.1142/S0219199718500633.

[15]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656.  doi: 10.1137/140985445.

[16]

J. Wu and Y. Wu, Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion, Adv. Math., 310 (2017), 759-888.  doi: 10.1016/j.aim.2017.02.013.

[17]

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 Sci., 25 (2015), 157-192.  doi: 10.1007/s00332-014-9224-7.

show all references

References:
[1]

D. Adhikar, C. Cao, J. Wu and X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Differential Equations, 256 (2014), 3594-3613.  doi: 10.1016/j.jde.2014.02.012.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

P. Constantin and C. R. Doering, Infinite Prandtl number convection, J. Stat. Phys., 94 (1999), 159-172.  doi: 10.1023/A:1004511312885.

[4]

T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-gepstrophic and inviscid boussinesq systems, SIAM. J. Math. Anal., 47 (2015), 4672-4684.  doi: 10.1137/14099036X.

[5] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982. 
[6]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier- Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[7]

C. 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), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[8]

A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lect. Notes Math., vol. 9, AMS/CIMS, 2003. doi: 10.1090/cln/009.

[9]

J. Pedlosky, Geoph Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

[10]

A. Pekalski and K. Sznajd-Weron (Eds.), Anomalous Diffusion, Form Basics to Applications, Lecture Notes in Phys., vol. 519, Springer-Verlag, Berlin, 1999.

[11]

X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020.

[12]

R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys., 67 (2016), Art. 104, 22 pp. doi: 10.1007/s00033-016-0697-0.

[13]

R. Wan, Global well-posedness of strong solutions to the 2D damped Boussinesq and MHD equations with large velocity, Comm. Math. Sci., 15 (2017), 1617-1626.  doi: 10.4310/CMS.2017.v15.n6.a6.

[14]

R. Wan, Long time stability for the dispersive SQG equation and Bousinessq equations in Sobolev space $H^s$, Commun. Contemp. Math., 2018. doi: 10.1142/S0219199718500633.

[15]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656.  doi: 10.1137/140985445.

[16]

J. Wu and Y. Wu, Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion, Adv. Math., 310 (2017), 759-888.  doi: 10.1016/j.aim.2017.02.013.

[17]

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 Sci., 25 (2015), 157-192.  doi: 10.1007/s00332-014-9224-7.

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