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

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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

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

Received July 2015 Revised September 2016 Published November 2016

Abstract
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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, (1975). Google Scholar
[2]	H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system,, J. Differential Equations, 233 (2007), 199. 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. 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. 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, (1980), 129. 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. 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. 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. 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. Google Scholar
[11]	B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics,, Differential Integral Equations, 11 (1998), 377. Google Scholar
[12]	G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. Google Scholar
[13]	E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-010-0357-z. Google Scholar
[24]	L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed.,, Pergamon, (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. 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. 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. doi: 10.1002/cpa.21506. Google Scholar
[28]	P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. I, II., Oxford University Press, (1996). 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. 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. 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. 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. doi: 10.1016/j.aim.2011.05.008. Google Scholar

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References:

[1]	R. A. Adams, Sobolev Spaces,, Academic, (1975). Google Scholar
[2]	H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system,, J. Differential Equations, 233 (2007), 199. 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. 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. 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, (1980), 129. 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. 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. 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. 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. Google Scholar
[11]	B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics,, Differential Integral Equations, 11 (1998), 377. Google Scholar
[12]	G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. Google Scholar
[13]	E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-010-0357-z. Google Scholar
[24]	L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed.,, Pergamon, (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. 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. 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. doi: 10.1002/cpa.21506. Google Scholar
[28]	P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. I, II., Oxford University Press, (1996). 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. 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. 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. 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. doi: 10.1016/j.aim.2011.05.008. Google Scholar

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