Abstract
In this paper, we show the global existence and uniqueness of strong and smooth large solutions to the 3D Boussinesq-MHD system without heat diffusion. Since the temperature satisfies a transport equation, in order to get high regularity of temperature, we need to use the combination of estimates about velocity and magnetic field. Moreover, our system involves a nonlinear damping term in the momentum equations due to the Brinkman–Forchheimer–extended-Darcy law of flow in porous media.
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Abidi, H., Hmidi, T.: On the global well-posedness for Boussinesq system. J. Differ. Equ. 233, 199–220 (2007)
Bian, D.: Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete Contin. Dyn. Syst. Ser. S 9(6), 1591–1611 (2016)
Bian, D., Gui, G.: On 2-D Boussinesq equations for MHD convection with stratification effects. J. Differ. Equ. 261, 1669–1711 (2016)
Bian, D., Gui, G., Guo, B., Xin, Z.: On the stability for the incompressible 2-D Boussinesq system for magnetohydrodynamics convection, preprint (2015)
Bian, D., Guo, B.: Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinet. Relat. Models 6(3), 481–503 (2013)
Bian, D., Liu, J.: Initial-boundary value problem to 2D Boussinesq equations for MHD convection with stratification effects. J. Differ. Equ. 263, 8074–8101 (2017)
Cai, Y., Lei, Z.: Global well-posedness of the incompressible magnetohydrodynamics. Arch. Rational Mech. Anal. 228, 969–993 (2018)
Cannon, J. R., Dibenedetto, E.: The initial value problem for the Boussinesq with data in \(L^p\). In: Approximation Methods for Navier–Stokes Problems, Lecture Notes in Mathematics, vol. 771, pp. 129–144. Springer, Berlin (1980)
Cao, C., Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226, 1803–1822 (2011)
Cao, C., Wu, J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)
Chae, D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203, 497–513 (2006)
Danchin, R., Paicu, M.: Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux. Bull. Soc. Math. France 136, 261–309 (2008)
Duvaut, G., Lions, J.L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)
Fang, D., Liu, C., Qian, C.: On partial regularity problem for 3D Boussinesq equations. J. Differ. Equ. 263, 4156–4221 (2017)
He, C., Xin, Z.: Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations. J. Funct. Anal. 227, 113–152 (2005)
He, C., Xin, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 213, 235–254 (2005)
He, L., Xu, L., Yu, P.: On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfven waves. Ann. PDE 4, 5 (2018)
Hmidi, T., Keraani, S.: On the global well-posedness of the Boussinesq system with zero viscosity. Indiana Univ. Math. J. 58, 1591–1618 (2009)
Hmidi, T., Rousset, F.: Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data. Ann. I. H. Poincare-AN. 27, 1227–1246 (2010)
Hmidi, T., Rousset, F.: Global well-posedness for the Euler–Boussinesq system with axisymmetric data. J. Funct. Anal. 260, 745–796 (2011)
Hou, T.Y., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12, 1–12 (2005)
Kang, K., Lee, J.: Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations. J. Differ. Equ. 247, 2310–2330 (2009)
Kulikovskiy, A.G., Lyubimov, G.A.: Magnetohydrodynamics. Addison-Wesley, Reading (1965)
Lai, M.J., Pan, R., Zhao, K.: Initial boundary value problem for two-dimensional viscous Boussinesq equations. Arch. Ration. Mech. Anal. 199, 739–760 (2011)
Laudau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 2nd edn. Pergamon, New York (1984)
Lei, Z.: On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Differ. Equ. 259, 3202–3215 (2015)
Li, D., Xu, X.: Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity. Dyn. PDE 10(3), 255–265 (2013)
Lin, F., Xu, L., Zhang, P.: Global small solutions of 2-D incompressible MHD system. arXiv:1302.5877v2
Lin, F., Zhang, P.: Global small solutions to an MHD-type system: the three-dimensional case. Commun. Pure Appl. Math. 67, 531–580 (2014)
Larios, A., Pei, Y.: On the local well-posedness and a Prodi–Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion. J. Differ. Equ. 263, 1419–1450 (2017)
Pratt, J., Busse, A., Müller, W.C.: Fluctuation dynamo amplified by intermittent shear bursts in convectively driven magnetohydrodynamic turbulence. Astron. Astrophys. 557, A76 (2013)
Ren, X., Wu, J., Xiang, Z., Zhang, Z.: Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J. Funct. Anal. 267, 503–541 (2014)
Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)
Taylor, M.: Partial Differential Equations III. Applied Mathematical Sciences, vol. 117. Springer, New York (1997)
Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam (1977)
Titi, E.S., Trabelsi, S.: Global well-posedness of a 3D MHD model in porous media. arXiv:1805.10661v2
Wang, C., Zhang, Z.: Global well-posedness for 2-D Boussinesq system with the temperature-density viscosity and thermal diffusivity. Adv. Math. 228, 43–62 (2011)
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The second author D. Bian was partially supported by NSFC under the contracts 11871005 and 11771041. The third author X. Pu was partially supported by NSFC under the contract 11871172.
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Liu, H., Bian, D. & Pu, X. Global well-posedness of the 3D Boussinesq-MHD system without heat diffusion. Z. Angew. Math. Phys. 70, 81 (2019). https://doi.org/10.1007/s00033-019-1126-y
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DOI: https://doi.org/10.1007/s00033-019-1126-y