Global existence of smooth solutions for three-dimensional magnetic Bénard system with mixed partial dissipation, magnetic diffusion and thermal diffusivity. (English) Zbl 1390.35277
Summary: This paper deals with the Cauchy problem to the 3D system of incompressible magnetic Bénard fluids. We prove that as the initial data satisfy \(\| u_0 \|_{H^1(\mathbb{R}^3)}^2 + \| b_0 \|_{H^1(\mathbb{R}^3)}^2 + \| \theta_0 \|_{H^1(\mathbb{R}^3)}^2 \leq \varepsilon\), where \(\varepsilon\) is a suitably small positive number, then the 3D magnetic Bénard system with mixed partial dissipation, magnetic diffusion and thermal diffusivity admit global smooth solutions.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76R10 | Free convection |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 78A30 | Electro- and magnetostatics |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
magnetic Bénard system; partial dissipation; magnetic diffusion; thermal diffusivity; smooth solutionReferences:
| [1] | Ambrosetti, A.; Prodi, G., A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0818.47059 |
| [2] | Bardos, C.; Degond, P., Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2, 101-118 (1985) · Zbl 0593.35076 |
| [3] | Benachour, S.; Filbet, F.; Laurençot, P.; Sonnendrücker, E., Global existence for the Vlasov-Darwin system in \(R^3\) for small initial data, Math. Methods Appl. Sci., 26, 297-319 (2003) · Zbl 1024.35027 |
| [4] | Cao, C.; Wu, J., Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226, 1803-1822 (2011) · Zbl 1213.35159 |
| [5] | Chae, D.; Huh, H., Global existence for small initial data in the Born-Infeld equations, J. Math. Phys., 44, 6132-6139 (2003) · Zbl 1063.81042 |
| [6] | Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics (1961), Clarendon Press: Clarendon Press Oxford · Zbl 0142.44103 |
| [7] | Charki, Z.; Zeytounian, R. Kh., The Bénard problem for deep convection: Lorenz deep system, Internat. J. Engrg. Sci., 32, 1561-1566 (1994) · Zbl 0899.76332 |
| [8] | Chemin, J.-Y., Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications, vol. 14 (1998), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie · Zbl 0927.76002 |
| [9] | Cheng, J.; Du, L., On two-dimensional magnetic Bénard problem with mixed partial viscosity, J. Math. Fluid Mech., 17, 769-797 (2015) · Zbl 1329.35245 |
| [10] | Du, L.; Zhou, D., Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal., 47, 1562-1589 (2015) · Zbl 1323.35143 |
| [11] | Guidoboni, G.; Padula, M., On the Bénard problem, (Trends in Partial Differential Equations of Mathematical Physics. Trends in Partial Differential Equations of Mathematical Physics, Progr. Nonlinear Differential Equations Appl. (2005), Birkhäuser: Birkhäuser Basel) · Zbl 1076.35097 |
| [12] | Kapustyan, A.; Pankov, A. V.; Valero, J., On global attractors of multivalued semiflows generated by the 3D Bénard system, Set-Valued Var. Anal., 20, 445-465 (2012) · Zbl 1259.35042 |
| [13] | Lin, H.; Du, L., Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions, Nonlinearity, 26, 219-239 (2013) · Zbl 1273.35076 |
| [14] | Lindblad, H., A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time, Proc. Amer. Math. Soc., 132, 1095-1102 (2004) · Zbl 1061.35053 |
| [15] | Ma, T.; Wang, S., Rayleigh-Bénard convection: dynamics and structure in the physical space, Commun. Math. Sci., 5, 553-574 (2007) · Zbl 1133.35426 |
| [16] | Miao, C., On the uniqueness theorems for the unbounded classical solution of the magnetic Bénard system, Systems Sci. Math. Sci., 13, 277-284 (2000) · Zbl 1050.76060 |
| [17] | Molinet, L.; Ribaud, F., Well-posedness results for the generalized Benjamin-Ono equation with small initial data, J. Math. Pures Appl., 83, 277-311 (2004) · Zbl 1084.35094 |
| [18] | Nakamura, M., Regularity and analyticity of the solutions of the magnetic Bénard problem, Adv. Math. Sci. Appl., 2, 117-137 (1993) · Zbl 0795.35086 |
| [19] | Ortega-Torres, E. E.; Rojas-Medar, M. A., Magneto-micropolar fluid motion: global existence of strong solutions, Abstr. Appl. Anal., 4, 109-125 (1999) · Zbl 0976.35055 |
| [20] | Rabinowitz, P. H., Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. Ration. Mech. Anal., 29, 179-235 (1968) · Zbl 0164.28704 |
| [21] | Rein, G.; Rendall, A. D., Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Comm. Math. Phys., 150, 561-583 (1992) · Zbl 0774.53056 |
| [22] | Seehafer, M., Global classical solutions of the Vlasov-Darwin system for small initial data, Commun. Math. Sci., 6, 749-764 (2008) · Zbl 1157.35335 |
| [23] | Wang, F.; Wang, K., Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14, 526-535 (2013) · Zbl 1253.35128 |
| [24] | Wang, Y.; Wang, K., Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity, Nonlinear Anal. Real World Appl., 33, 348-362 (2017) · Zbl 1352.35126 |
| [25] | Wu, G.; Xue, L., Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and Yudovich’s type data, J. Differential Equations, 253, 100-125 (2012) · Zbl 1305.35119 |
| [26] | Zhou, Y.; Fan, J.; Nakamura, G., Global Cauchy problem for a 2D magnetic Bénard problem with zero thermal conductivity, Appl. Math. Lett., 26, 627-630 (2013) · Zbl 1355.35155 |
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