December  2016, 9(6): 1591-1611. doi: 10.3934/dcdss.2016065

Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Received  July 2015 Revised  September 2016 Published  November 2016

This paper is concerned with the initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. We show that the system has a unique classical solution for $H^3$ initial data, and the non-slip boundary condition for velocity field and the perfectly conducting wall condition for magnetic field. In addition, we show that the kinetic energy is uniformly bounded in time.
Citation: Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065
References:
[1]

R. A. Adams, Sobolev Spaces, Academic, New York, 1975.

[2]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations, 233 (2007), 199-220. doi: 10.1016/j.jde.2006.10.008.

[3]

D. Bian and G. Gui, On 2-D Boussinesq equations for MHD convection with stratification effects, J. Differential Equations, 261 (2016), 1669-1711. doi: 10.1016/j.jde.2016.04.011.

[4]

D. Bian, G. Gui, B. Guo and Z. Xin, On the stability for the incompressible 2-D Boussinesq system for magnetohydrodynamics convection, preprint, 2015.

[5]

D. Bian and B. Guo, Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations, Kinetic and Related Models, 6 (2013), 481-503. doi: 10.3934/krm.2013.6.481.

[6]

J. R. Cannon and E. Dibenedetto, The initial value problem for the Boussinesqs with data in $L^p$. In: Approximation Methods for Navier-Stokes Problems, Lecture Notes in Mathematics Volume, vol. 771, pp. 129-144. Springer, Berlin, 1980.

[7]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[8]

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.

[9]

Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magnetohydrodynamics equations, Comm. Math. Phys., 275 (2007), 861-872. doi: 10.1007/s00220-007-0319-y.

[10]

R. Danchin and M. Paicu, Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136 (2008), 261-309.

[11]

B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394.

[12]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.

[13]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.

[14]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.

[15]

J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452.

[16]

G. Gui, Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system withvariable density and electrical conductivity, J. Functional Analysis, 267 (2014), 1488-1539. doi: 10.1016/j.jfa.2014.06.002.

[17]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Functional Analysis, 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.

[18]

T. Hmidi and S. Keraani, 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.

[19]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. I. H. Poincare-AN., 27 (2010), 1227-1246. doi: 10.1016/j.anihpc.2010.06.001.

[20]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Functional Analysis, 260 (2011), 745-796. doi: 10.1016/j.jfa.2010.10.012.

[21]

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

[22]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1968.

[23]

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.

[24]

L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Pergamon, New York, 1984.

[25]

D. Li and X. Xu, Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ., 10 (2013), 255-265. doi: 10.4310/DPDE.2013.v10.n3.a2.

[26]

F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485, arXiv: 1302.5877v2. doi: 10.1016/j.jde.2015.06.034.

[27]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.

[28]

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. I, II. Oxford University Press, New York, 1996, 1998.

[29]

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. Functional Analysis, 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020.

[30]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations}, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[31]

W. von Wahl, Estimating $\nabla u$ by divu and curlu, Math. Methods Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206.

[32]

C. Wang and Z. Zhang, Global well-posedness for 2-D Boussinesq system with the temperature-density viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic, New York, 1975.

[2]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations, 233 (2007), 199-220. doi: 10.1016/j.jde.2006.10.008.

[3]

D. Bian and G. Gui, On 2-D Boussinesq equations for MHD convection with stratification effects, J. Differential Equations, 261 (2016), 1669-1711. doi: 10.1016/j.jde.2016.04.011.

[4]

D. Bian, G. Gui, B. Guo and Z. Xin, On the stability for the incompressible 2-D Boussinesq system for magnetohydrodynamics convection, preprint, 2015.

[5]

D. Bian and B. Guo, Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations, Kinetic and Related Models, 6 (2013), 481-503. doi: 10.3934/krm.2013.6.481.

[6]

J. R. Cannon and E. Dibenedetto, The initial value problem for the Boussinesqs with data in $L^p$. In: Approximation Methods for Navier-Stokes Problems, Lecture Notes in Mathematics Volume, vol. 771, pp. 129-144. Springer, Berlin, 1980.

[7]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[8]

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.

[9]

Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magnetohydrodynamics equations, Comm. Math. Phys., 275 (2007), 861-872. doi: 10.1007/s00220-007-0319-y.

[10]

R. Danchin and M. Paicu, Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136 (2008), 261-309.

[11]

B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394.

[12]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.

[13]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.

[14]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.

[15]

J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452.

[16]

G. Gui, Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system withvariable density and electrical conductivity, J. Functional Analysis, 267 (2014), 1488-1539. doi: 10.1016/j.jfa.2014.06.002.

[17]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Functional Analysis, 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.

[18]

T. Hmidi and S. Keraani, 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.

[19]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. I. H. Poincare-AN., 27 (2010), 1227-1246. doi: 10.1016/j.anihpc.2010.06.001.

[20]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Functional Analysis, 260 (2011), 745-796. doi: 10.1016/j.jfa.2010.10.012.

[21]

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

[22]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1968.

[23]

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.

[24]

L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Pergamon, New York, 1984.

[25]

D. Li and X. Xu, Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ., 10 (2013), 255-265. doi: 10.4310/DPDE.2013.v10.n3.a2.

[26]

F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485, arXiv: 1302.5877v2. doi: 10.1016/j.jde.2015.06.034.

[27]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.

[28]

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. I, II. Oxford University Press, New York, 1996, 1998.

[29]

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. Functional Analysis, 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020.

[30]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations}, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[31]

W. von Wahl, Estimating $\nabla u$ by divu and curlu, Math. Methods Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206.

[32]

C. Wang and Z. Zhang, Global well-posedness for 2-D Boussinesq system with the temperature-density viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.

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