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

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

Dongfen Bian 1,

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.
Keywords: MHD Boussinesq system, global regularity, weak solution, uniqueness..
Mathematics Subject Classification: Primary: 35Q35, 76D0.
Citation: Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

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