# American Institute of Mathematical Sciences

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:
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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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