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 & 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). Google Scholar

[2]

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

[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. Google Scholar

[4]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms,, Adv. Math., 203 (2006), 497. doi: 10.1016/j.aim.2005.05.001. Google Scholar

[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. doi: 10.1088/0951-7715/19/11/006. Google Scholar

[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. doi: 10.1017/S0308210500026810. Google Scholar

[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. doi: 10.1007/s00220-009-0821-5. Google Scholar

[8]

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

[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. doi: 10.1016/0362-546X(87)90061-7. Google Scholar

[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. Google Scholar

[11]

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

[12]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems. Vol. 1B.,, Elsevier B. V., (2006). Google Scholar

[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. Google Scholar

[14]

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

[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. doi: 10.1080/03605302.2010.518657. Google Scholar

[16]

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

[17]

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

[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. doi: 10.1016/j.na.2014.10.030. Google Scholar

[19]

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

[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. doi: 10.1137/140958256. Google Scholar

[21]

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

[22]

T. Kato and G. Ponce, Commutator estimates and the euler and navier-stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704. Google Scholar

[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. doi: 10.1016/j.physd.2010.11.012. Google Scholar

[24]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Lezioni Lincee. [Lincei Lectures], (1991). doi: 10.1017/CBO9780511569418. Google Scholar

[25]

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

[26]

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

[27]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. III,, Springer-Verlag, (1972). Google Scholar

[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. doi: 10.1142/S0219891615500137. Google Scholar

[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. doi: 10.1007/s00205-010-0357-z. Google Scholar

[30]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, in Cambridge Texts in Applied Mathematics, (2002). Google Scholar

[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. Google Scholar

[32]

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

[33]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer Verlag, (1987). Google Scholar

[34]

A. Stefanov and J. Wu, A gloval regularity result for the 2D Boussinesq equations with critical dissipation, preprint,, , (). Google Scholar

[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. doi: 10.1016/j.jde.2013.04.032. Google Scholar

[36]

R. Temam, Navier-Stokes Equations,, $3^{rd}$ edition, (1984). Google Scholar

[37]

________, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1988). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[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. doi: 10.1090/conm/371/06862. Google Scholar

[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. doi: 10.1002/cpa.20170. Google Scholar

[40]

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

[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. doi: 10.1016/j.jde.2012.02.025. Google Scholar

[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. doi: 10.1016/j.jde.2014.01.038. Google Scholar

[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. doi: 10.1016/j.jde.2014.08.006. Google Scholar

[44]

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

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[4]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms,, Adv. Math., 203 (2006), 497. doi: 10.1016/j.aim.2005.05.001. Google Scholar

[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. doi: 10.1088/0951-7715/19/11/006. Google Scholar

[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. doi: 10.1017/S0308210500026810. Google Scholar

[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. doi: 10.1007/s00220-009-0821-5. Google Scholar

[8]

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

[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. doi: 10.1016/0362-546X(87)90061-7. Google Scholar

[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. Google Scholar

[11]

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

[12]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems. Vol. 1B.,, Elsevier B. V., (2006). Google Scholar

[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. Google Scholar

[14]

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

[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. doi: 10.1080/03605302.2010.518657. Google Scholar

[16]

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

[17]

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

[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. doi: 10.1016/j.na.2014.10.030. Google Scholar

[19]

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

[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. doi: 10.1137/140958256. Google Scholar

[21]

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

[22]

T. Kato and G. Ponce, Commutator estimates and the euler and navier-stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704. Google Scholar

[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. doi: 10.1016/j.physd.2010.11.012. Google Scholar

[24]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Lezioni Lincee. [Lincei Lectures], (1991). doi: 10.1017/CBO9780511569418. Google Scholar

[25]

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

[26]

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

[27]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. III,, Springer-Verlag, (1972). Google Scholar

[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. doi: 10.1142/S0219891615500137. Google Scholar

[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. doi: 10.1007/s00205-010-0357-z. Google Scholar

[30]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, in Cambridge Texts in Applied Mathematics, (2002). Google Scholar

[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. Google Scholar

[32]

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

[33]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer Verlag, (1987). Google Scholar

[34]

A. Stefanov and J. Wu, A gloval regularity result for the 2D Boussinesq equations with critical dissipation, preprint,, , (). Google Scholar

[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. doi: 10.1016/j.jde.2013.04.032. Google Scholar

[36]

R. Temam, Navier-Stokes Equations,, $3^{rd}$ edition, (1984). Google Scholar

[37]

________, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1988). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[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. doi: 10.1090/conm/371/06862. Google Scholar

[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. doi: 10.1002/cpa.20170. Google Scholar

[40]

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

[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. doi: 10.1016/j.jde.2012.02.025. Google Scholar

[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. doi: 10.1016/j.jde.2014.01.038. Google Scholar

[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. doi: 10.1016/j.jde.2014.08.006. Google Scholar

[44]

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

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