Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T00:33:37.860Z Has data issue: false hasContentIssue false
  • English
  • Français

Global well-posedness and decay estimates of strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection

Part of: Partial differential equations Equations of mathematical physics and other areas of application Incompressible viscous fluids

Published online by Cambridge University Press:  15 September 2020

Xin Zhong
Xin Zhong*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing400715, People's Republic of China (xzhong1014@amss.ac.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

We deal with an initial boundary value problem of nonhomogeneous Boussinesq equations for magnetohydrodynamics convection in two-dimensional domains. We prove that there is a unique global strong solution. Moreover, we show that the temperature converges exponentially to zero in H1 as time goes to infinity. In particular, the initial data can be arbitrarily large and vacuum is allowed. Our analysis relies on energy method and a lemma of Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).

Keywords

Nonhomogeneous Boussinesq-MHD equationsglobal well-posednessdecay estimateslarge initial datavacuum

MSC classification

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 35Q35: PDEs in connection with fluid mechanics 76D03: Existence, uniqueness, and regularity theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1543 - 1567
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was partially supported by National Natural Science Foundation of China (No. 11901474).

References

Amrouche, C. and Girault, V.. Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math J. 44 (1994), 109–140.CrossRefGoogle Scholar
Bian, D.. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete Contin. Dyn. Syst. Ser. S. 9 (2016), 1591–1611.CrossRefGoogle Scholar
Bian, D. and Gui, G.. On 2-D Boussinesq equations for MHD convection with stratification effects. J. Diff. Equ. 261 (2016), 1669–1711.CrossRefGoogle Scholar
Bian, D. and Liu, J.. Initial-boundary value problem to 2D Boussinesq equations for MHD convection with stratification effects. J. Diff. Equ. 263 (2017), 8074–8101.CrossRefGoogle Scholar
Bian, D. and Pu, X.. Global smooth axisymmetric solutions of the Boussinesq equations for magnetohydrodynamics convection. J. Math. Fluid Mech. 22 (2020), Paper No. 12.CrossRefGoogle Scholar
Cao, C. and Wu, J.. Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. Arch. Rational Mech. Anal. 208 (2013), 985–1004.CrossRefGoogle Scholar
Chae, D.. Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203 (2006), 497–513.CrossRefGoogle Scholar
Desjardins, B.. Regularity results for two-dimensional flows of multiphase viscous fluids. Arch. Rational Mech. Anal. 137 (1997), 135–158.CrossRefGoogle Scholar
Evans, L. C.. Partial differential equations, 2nd edn. (Providence, RI: American Mathematical Society, 2010).Google Scholar
Fan, J. and Ozawa, T.. Regularity criteria for the 3D density-dependent Boussinesq equations. Nonlinearity 22 (2009), 553–568.CrossRefGoogle Scholar
Fan, J., Li, F. and Nakamura, G.. Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum. Commun. Pure Appl. Anal. 13 (2014), 1481–1490.CrossRefGoogle Scholar
Friedman, A.. Partial differential equations (New York: Dover Books on Mathematics, 2008).Google Scholar
Lai, M.-J., Pan, R. and Zhao, K.. Initial boundary value problem for two-dimensional viscous Boussinesq equations. Arch. Ration. Mech. Anal. 199 (2011), 739–760.CrossRefGoogle Scholar
Larios, A. and 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. Diff. Equ. 263 (2017), 1419–1450.CrossRefGoogle Scholar
Li, J.. Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density. J. Diff. Equ. 263 (2017), 6512–6536.CrossRefGoogle Scholar
Li, J. and Titi, E. S.. Global well-posedness of the 2D Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal. 220 (2016), 983–1001.CrossRefGoogle Scholar
Lions, P. L.. Mathematical topics in fluid mechanics, vol. I incompressible models (Oxford: Oxford University Press, 1996).Google Scholar
Liu, H., Bian, D. and Pu, X.. Global well-posedness of the 3D Boussinesq-MHD system without heat diffusion. Z. Angew. Math. Phys. 70 (2019), Paper No. 81.CrossRefGoogle Scholar
Majda, A.. Introduction to PDEs and waves for the atmosphere and ocean (Providence, RI: American Mathematical Society, 2003).CrossRefGoogle Scholar
Pedlosky, J.. Geophysical fluid dynamics (New York: Springer-Verlag, 1987).CrossRefGoogle Scholar
Qiu, H. and Yao, Z.. Well-posedness for density-dependent Boussinesq equations without dissipation terms in Besov spaces. Comput. Math. Appl. 73 (2017), 1920–1931.CrossRefGoogle Scholar
Song, S.. On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum. Z. Angew. Math. Phys. 69 (2018), Paper No. 23.CrossRefGoogle Scholar
Struwe, M.. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, 4th edn. (Berlin: Springer-Verlag, 2008).Google Scholar
Zhai, X. and Chen, Z. M.. Global well-posedness for the MHD-Boussinesq system with the temperature-dependent viscosity. Nonlinear Anal. Real World Appl. 44 (2018), 260–282.CrossRefGoogle Scholar
Zhang, Z.. 3D density-dependent Boussinesq equations with velocity field in BMO spaces. Acta Appl. Math. 142 (2016), 1–8.CrossRefGoogle Scholar
Zhong, X.. Global well-posedness to the Cauchy problem of two-dimensional density-dependent Boussinesq equations with large initial data and vacuum. Discrete Contin. Dyn. Syst. 39 (2019), 6713–6745.CrossRefGoogle Scholar
Zhong, X.. Strong solutions to the 2D Cauchy problem of density-dependent viscous Boussinesq equations with vacuum. J. Math. Phys. 60 (2019), 051505.CrossRefGoogle Scholar
Zhong, X.. Global strong solution and exponential decay for nonhomogeneous magnetohydrodynamic equations. Discrete Contin. Dyn. Syst. Ser. B. doi:10.3934/dcdsb.2020246.Google Scholar
Zhong, X.. Global well-posedness to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection with zero heat diffusion and large initial data, submitted for publication.Google Scholar