October  2016, 21(8): 2531-2550. doi: 10.3934/dcdsb.2016059

The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case

1. 

The Department of Mathematics, Indiana University, 831 East Third Street, Rawles Hall, Bloomington, Indiana 47405, United States

2. 

The Institute for Scienti c Computing and Applied Mathematics, Indiana University, 831 East Third Street, Rawles Hall, Bloomington, Indiana 47405, United States

Received  August 2015 Revised  April 2016 Published  September 2016

We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature.
Citation: Wenru Huo, Aimin Huang. The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2531-2550. doi: 10.3934/dcdsb.2016059
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 1992.

[2]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.

[3]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, Approximation methods for Navier-Stokes problems Lecture Notes in Math., 771 (1980), 129-144.

[4]

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.

[5]

P. Constantin, M. Lewicka and L. Ryzhik, Travelling waves in two-dimensional reactive Boussinesq systems with no-slip boundary conditions, Nonlinearity, 19 (2006), 2605-2615. doi: 10.1088/0951-7715/19/11/006.

[6]

D. Chae and H.-S. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946. doi: 10.1017/S0308210500026810.

[7]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14. doi: 10.1007/s00220-009-0821-5.

[8]

W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58. doi: 10.1063/1.868044.

[9]

C. Foias, O. Manley and R. Temam, Attractors for the Bénard problem: Existence and physical bounds on their fractal dimension, Nonlinear Anal., 11 (1987), 939-967. doi: 10.1016/0362-546X(87)90061-7.

[10]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[11]

S. Gatti, V. Pata and S. Zelik, A gronwall-type lemma with parameter and dissipative estimates for PDEs, Nonlinear Anal., 70 (2009), 2337-2343. doi: 10.1016/j.na.2008.03.015.

[12]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems. Vol. 1B., Elsevier B. V., Amsterdam, 2006.

[13]

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.

[14]

_______, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618. doi: 10.1512/iumj.2009.58.3590.

[15]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445. doi: 10.1080/03605302.2010.518657.

[16]

W. Hu, I. Kukavica and M. Ziane, On the regularity for the Boussinesq equations in a bounded domain, J. Math. Phys., 54 (2013), 081507, 10 pp. doi: 10.1063/1.4817595.

[17]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.

[18]

A. Huang, The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity, Nonlinear Anal., 113 (2015), 401-429. doi: 10.1016/j.na.2014.10.030.

[19]

_______, The 2d Euler-Boussinesq equations in planar polygonal domains with Yudovich's type data, Commun. Math. Stat., 2 (2014), 369-391. doi: 10.1007/s40304-015-0045-2.

[20]

Q. Jiu, C. Miao, J. Wu and Z. Zhang, The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454. doi: 10.1137/140958256.

[21]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161-181. doi: 10.1007/s00220-004-1256-7.

[22]

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.

[23]

J. P. Kelliher, R. Temam and X. Wang, Boundary layer associated with the Darcy-Brinkman-Boussinesq model for convection in porous media, Phys. D, 240 (2011), 619-628. doi: 10.1016/j.physd.2010.11.012.

[24]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. (92k:58040) doi: 10.1017/CBO9780511569418.

[25]

S. A. Lorca and J. L. Boldrini, Stationary solutions for generalized Boussinesq models, J. Differential Equations, 124 (1996), 389-406. doi: 10.1006/jdeq.1996.0016.

[26]

_______, The initial value problem for a generalized boussinesq model, Nonlinear Anal., 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4.

[27]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. III, Springer-Verlag, New York-Heidelberg, 1972.

[28]

H. Li, R. Pan and W. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent heat diffusion, J. Hyperbolic Differ. Equ., 12 (2015), 469-488. doi: 10.1142/S0219891615500137.

[29]

M.-J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/s00205-010-0357-z.

[30]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, in Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.

[31]

A. Miranville and M. Ziane, On the dimension of the attractor for the Bénard problem with free surfaces, Russian J. Math. Phys., 5 (1997), 489-502.

[32]

V. Pata, Uniform estimates of gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270. doi: 10.1016/j.jmaa.2010.07.006.

[33]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, Berlin, 1987.

[34]

A. Stefanov and J. Wu, A gloval regularity result for the 2D Boussinesq equations with critical dissipation, preprint, arXiv:1411.1362v1.

[35]

A. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-d Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.

[36]

R. Temam, Navier-Stokes Equations, $3^{rd}$ edition, North-Holland Publishing Co., Amsterdam, 1984.

[37]

________, Infinite-dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3.

[38]

X. Wang, A note on long time behavior of solutions to the Boussinesq system at large Prandtl number, Nonlinear partial differential equations and related analysis, 371 (2005), 315-323. doi: 10.1090/conm/371/06862.

[39]

________, Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number, Comm. Pure Appl. Math., 60 (2007), 1293-1318. doi: 10.1002/cpa.20170.

[40]

J. Wu, The quasi-geostrophic equation and its two regularizations, Comm. Partial Differential Equations, 27 (2002), 1161-1181. doi: 10.1081/PDE-120004898.

[41]

G. Wu and L. Xue, Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and Yudovich's type data, J. Differential Equations, 253 (2012), 100-125. doi: 10.1016/j.jde.2012.02.025.

[42]

X. Xu and L. Xue, Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation, J. Differential Equations, 256 (2014), 3179-3207. doi: 10.1016/j.jde.2014.01.038.

[43]

W. Yang, Q. Jiu and J. Wu, Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, J. Differential Equations, 257 (2014), 4188-4213. doi: 10.1016/j.jde.2014.08.006.

[44]

K. Zhao, 2D inviscid heat conductive Boussinesq equations on a bounded domain, Michigan Math. J., 59 (2010), 329-352. doi: 10.1307/mmj/1281531460.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 1992.

[2]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.

[3]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, Approximation methods for Navier-Stokes problems Lecture Notes in Math., 771 (1980), 129-144.

[4]

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.

[5]

P. Constantin, M. Lewicka and L. Ryzhik, Travelling waves in two-dimensional reactive Boussinesq systems with no-slip boundary conditions, Nonlinearity, 19 (2006), 2605-2615. doi: 10.1088/0951-7715/19/11/006.

[6]

D. Chae and H.-S. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946. doi: 10.1017/S0308210500026810.

[7]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14. doi: 10.1007/s00220-009-0821-5.

[8]

W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58. doi: 10.1063/1.868044.

[9]

C. Foias, O. Manley and R. Temam, Attractors for the Bénard problem: Existence and physical bounds on their fractal dimension, Nonlinear Anal., 11 (1987), 939-967. doi: 10.1016/0362-546X(87)90061-7.

[10]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[11]

S. Gatti, V. Pata and S. Zelik, A gronwall-type lemma with parameter and dissipative estimates for PDEs, Nonlinear Anal., 70 (2009), 2337-2343. doi: 10.1016/j.na.2008.03.015.

[12]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems. Vol. 1B., Elsevier B. V., Amsterdam, 2006.

[13]

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.

[14]

_______, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618. doi: 10.1512/iumj.2009.58.3590.

[15]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445. doi: 10.1080/03605302.2010.518657.

[16]

W. Hu, I. Kukavica and M. Ziane, On the regularity for the Boussinesq equations in a bounded domain, J. Math. Phys., 54 (2013), 081507, 10 pp. doi: 10.1063/1.4817595.

[17]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.

[18]

A. Huang, The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity, Nonlinear Anal., 113 (2015), 401-429. doi: 10.1016/j.na.2014.10.030.

[19]

_______, The 2d Euler-Boussinesq equations in planar polygonal domains with Yudovich's type data, Commun. Math. Stat., 2 (2014), 369-391. doi: 10.1007/s40304-015-0045-2.

[20]

Q. Jiu, C. Miao, J. Wu and Z. Zhang, The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454. doi: 10.1137/140958256.

[21]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161-181. doi: 10.1007/s00220-004-1256-7.

[22]

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.

[23]

J. P. Kelliher, R. Temam and X. Wang, Boundary layer associated with the Darcy-Brinkman-Boussinesq model for convection in porous media, Phys. D, 240 (2011), 619-628. doi: 10.1016/j.physd.2010.11.012.

[24]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. (92k:58040) doi: 10.1017/CBO9780511569418.

[25]

S. A. Lorca and J. L. Boldrini, Stationary solutions for generalized Boussinesq models, J. Differential Equations, 124 (1996), 389-406. doi: 10.1006/jdeq.1996.0016.

[26]

_______, The initial value problem for a generalized boussinesq model, Nonlinear Anal., 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4.

[27]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. III, Springer-Verlag, New York-Heidelberg, 1972.

[28]

H. Li, R. Pan and W. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent heat diffusion, J. Hyperbolic Differ. Equ., 12 (2015), 469-488. doi: 10.1142/S0219891615500137.

[29]

M.-J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/s00205-010-0357-z.

[30]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, in Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.

[31]

A. Miranville and M. Ziane, On the dimension of the attractor for the Bénard problem with free surfaces, Russian J. Math. Phys., 5 (1997), 489-502.

[32]

V. Pata, Uniform estimates of gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270. doi: 10.1016/j.jmaa.2010.07.006.

[33]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, Berlin, 1987.

[34]

A. Stefanov and J. Wu, A gloval regularity result for the 2D Boussinesq equations with critical dissipation, preprint, arXiv:1411.1362v1.

[35]

A. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-d Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.

[36]

R. Temam, Navier-Stokes Equations, $3^{rd}$ edition, North-Holland Publishing Co., Amsterdam, 1984.

[37]

________, Infinite-dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3.

[38]

X. Wang, A note on long time behavior of solutions to the Boussinesq system at large Prandtl number, Nonlinear partial differential equations and related analysis, 371 (2005), 315-323. doi: 10.1090/conm/371/06862.

[39]

________, Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number, Comm. Pure Appl. Math., 60 (2007), 1293-1318. doi: 10.1002/cpa.20170.

[40]

J. Wu, The quasi-geostrophic equation and its two regularizations, Comm. Partial Differential Equations, 27 (2002), 1161-1181. doi: 10.1081/PDE-120004898.

[41]

G. Wu and L. Xue, Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and Yudovich's type data, J. Differential Equations, 253 (2012), 100-125. doi: 10.1016/j.jde.2012.02.025.

[42]

X. Xu and L. Xue, Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation, J. Differential Equations, 256 (2014), 3179-3207. doi: 10.1016/j.jde.2014.01.038.

[43]

W. Yang, Q. Jiu and J. Wu, Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, J. Differential Equations, 257 (2014), 4188-4213. doi: 10.1016/j.jde.2014.08.006.

[44]

K. Zhao, 2D inviscid heat conductive Boussinesq equations on a bounded domain, Michigan Math. J., 59 (2010), 329-352. doi: 10.1307/mmj/1281531460.

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