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On well-posedness of vector-valued fractional differential-difference equations
Global well-posedness for the 2D Boussinesq equations with a velocity damping term
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China |
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 [
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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.
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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. |
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P. Constantin and C. R. Doering,
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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. |
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A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982.
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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.
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J. Pedlosky, Geoph Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. |
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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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