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

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

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 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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