Advances in Differential Equations

Global regularity of the 2D magnetic Bénard system with partial dissipation

Zhuan Ye

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 23, Number 3/4 (2018), 193-238.

Dates
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1513652446

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics 35B65: Smoothness and regularity of solutions 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

Citation

Ye, Zhuan. Global regularity of the 2D magnetic Bénard system with partial dissipation. Adv. Differential Equations 23 (2018), no. 3/4, 193--238. https://projecteuclid.org/euclid.ade/1513652446


Export citation