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March/April 2018 Global regularity of the 2D magnetic Bénard system with partial dissipation
Zhuan Ye
Adv. Differential Equations 23(3/4): 193-238 (March/April 2018).

Abstract

In this paper, we consider the Cauchy problem of the two-dimensional (2D) magnetic Bénard system with partial dissipation. On the one hand, we obtain the global regularity of the 2D magnetic Bénard system with zero thermal conductivity. The main difficulty is the zero thermal conductivity. To bypass this difficulty, we exploit the structure of the coupling system about the vorticity and the temperature and use the Maximal $L_t^{p}L_x^{q}$ regularity for the heat kernel. On the other hand, we also establish the global regularity of the 2D magnetic Bénard system with horizontal dissipation, horizontal magnetic diffusion and with either horizontal or vertical thermal diffusivity. This settles the global regularity issue unsolved in the previous works. Additionally, in the Appendix, we also show that with a full Laplacian for the diffusive term of the magnetic field and half of the full Laplacian for the temperature field, the global regularity result holds true as long as the power $\alpha$ of the fractional Laplacian dissipation for the velocity field is positive.

Citation

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Zhuan Ye. "Global regularity of the 2D magnetic Bénard system with partial dissipation." Adv. Differential Equations 23 (3/4) 193 - 238, March/April 2018.

Information

Published: March/April 2018
First available in Project Euclid: 19 December 2017

zbMATH: 06822198
MathSciNet: MR3738646

Subjects:
Primary: 35B65, 35Q35, 76D03

Rights: Copyright © 2018 Khayyam Publishing, Inc.

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Vol.23 • No. 3/4 • March/April 2018
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