×

On the infinite Prandtl number limit in two-dimensional magneto-convection. (English) Zbl 1412.35275

Summary: In this paper, the infinite limit of the Prandtl number is justified for the two-dimensional incompressible magneto-convection, which describes the nonlinear interaction between the Rayleigh-Bénard convection and an externally magnetic field. Both the convergence rates and the thickness of initial layer are obtained. Moreover, based on the method of formal asymptotic expansions, an effective dynamics is constructed to simulate the motion within the initial layer.

MSC:

35Q35 PDEs in connection with fluid mechanics
35C20 Asymptotic expansions of solutions to PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
76R10 Free convection
35Q85 PDEs in connection with astronomy and astrophysics
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Davidson, P. A., An Introduction to Magnetohydrodynamics (2001), Combridge University Press · Zbl 0974.76002
[2] Busse, F. H., Fundamentals of Thermal Convection. Mantel convection: plate tectonics and fluid dynamics, (Peltier, W. R., The Fluid Mechanics of Astrophysics and Geophysics, Vol. 4 (1989), Gordon and Breach: Gordon and Breach New York), 23-95
[3] Getling, A. V., (Rayleigh-Bénard Convection. Structure and Dynamics. Rayleigh-Bénard Convection. Structure and Dynamics, Advanced Series in Nonlinear Dynamics, vol. 11 (1998), World Scientific: World Scientific River Edge, N.J.) · Zbl 0910.76001
[4] Chandrasekhar, S., On the inhibition of convection by a magnetic field, Phil. Mag. Ser. 7, 43, 501-532 (1952) · Zbl 0046.24002
[5] Chandrasekhar, S., On the inhibition of convection by a magnetic field. II, Phil. Mag. Ser. 7, 45, 1177-1191 (1954) · Zbl 0056.43802
[6] Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability (1961), Clarendon: Clarendon Oxford · Zbl 0142.44103
[7] Nakagawa, Y., An experiment on the inhibition of thermal convection by a magnetic field, Nature, 175, 417-419 (1955)
[8] Nakagawa, Y., Experiments on the inhibition of thermal convection by a magnetic field, Proc. R. Soc. Lond. Ser. A, 240, 108-113 (1957)
[9] Parker, E. N., Cosmical Magnetic Fields (1979), Clarendon: Clarendon Oxford
[10] Thompson, W. B., Thermal convection in a magnetic field, Phil. Mag. Ser. 7, 42, 1417-1432 (1951) · Zbl 0043.45302
[11] Grossmann, S.; Lohse, D., Scaling in thermal convection: a unifying theory, J. Fluid Mech., 407, 27-56 (2000) · Zbl 0972.76045
[12] Siggia, E. D., High Rayleigh number convection, (Annual Review of Fluid Mechanics, Vol. 26 (1994), Annual Reviews: Annual Reviews Palo Alto, Calif.), 137-168 · Zbl 0800.76425
[13] Wang, X. M., Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure Appl. Math., 57, 10, 1265-1282 (2004) · Zbl 1112.76032
[14] Wang, X. M., Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number, Comm. Pure Appl. Math., 60, 9, 1293-1318 (2007) · Zbl 1143.35087
[15] Constantin, P.; Doering, C. R., Infinite Prandtl number convection, J. Stat. Phys., 94, 1-2, 159-172 (1999) · Zbl 0935.76083
[16] Constantin, P.; Hallstrom, C.; Poutkaradze, V., Logarithmic bounds for infinite Prandtl number rotating convection, J. Math. Phys., 42, 773-783 (2001) · Zbl 1061.76519
[17] Doering, C. R.; Constantin, P., On upper bounds for infinite Prandtl number convection with or without rotation, J. Math. Phys., 42, 784-795 (2001) · Zbl 1061.76520
[18] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow (1963), Silverman Gordon and Breach Science Publishers: Silverman Gordon and Breach Science Publishers New York-London · Zbl 0121.42701
[19] Temam, R., Navier-Stokes equations, (Theory and Numerical Analysis (2001), AMS: AMS Providence, R. I.) · Zbl 0994.35002
[20] Galdi, G. P., Nonlinear stability of the magnetic Bénard problem via a generalized energy method, Arch. Ration. Mech. Anal., 87, 2, 167-186 (1985) · Zbl 0611.76069
[21] Mulone, G.; Rionero, S., On the stability of the rotating Bénard problem, Bull. Tech. Univ. Istanb., 47, 181-202 (1994) · Zbl 0864.76030
[22] Mulone, G.; Rionero, S., Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem, Arch. Ration. Mech. Anal., 166, 3, 197-218 (2003) · Zbl 1022.76020
[23] Rionero, S., On magnetohydrodynamic stability, Quad. Mat., 1, 347-376 (1997) · Zbl 0960.76031
[24] Sermange, M.; Temam, R., Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36, 635-664 (1983) · Zbl 0524.76099
[25] Brenier, Y., Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation, Comm. Math. Phys., 330, 2, 757-770 (2014) · Zbl 1294.35077
[26] Mccormick, D. S.; Robinson, J. C.; Rodrigo, J. L., Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation, Arch. Ration. Mech. Anal., 214, 503-523 (2014) · Zbl 1317.35206
[27] Cao, C. S.; Wu, J. H., Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226, 2, 1803-1822 (2011) · Zbl 1213.35159
[28] Cao, C. S.; Wu, J. H.; Yuan, B. Q., The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46, 1, 588-602 (2014) · Zbl 1293.35233
[29] Holmes, M. H., Introduction to Perturbation Methods (1995), Springer: Springer New York · Zbl 0830.34001
[30] Majda, A., (Introduction to PDEs and Waves for the Atmosphere and Ocean. Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics (2003), AMS: AMS Providence, R. I.) · Zbl 1278.76004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.