Blow-up criteria for the 2½D magnetic Bénard fluid system with partial viscosity

ABSTRACT

We study the Cauchy problem for the 2$\frac{1}{2}$D incompressible magnetic Bénard fluid system with partial viscosity and derive some blow-up criteria of smooth solutions. More precisely, we first obtain a blow-up criterion of smooth solutions to the 2$\frac{1}{2}$D magnetic Bénard fluid system without zero magnetic diffusion and thermal diffusivity. Furthermore, we derive two blow-up criteria of smooth solutions to the 2$\frac{1}{2}$D magnetic Bénard fluid system without dissipation.

COMMUNICATED BY:

• Ming Mei

KEYWORDS:

• Magnetic Bénard system
• blow-up criterion
• smooth solutions
• dissipation
• magnetic diffusion
• thermal diffusivity
• Besov spaces

AMS CLASSIFICATIONS:

• 35B65
• 35Q35
• 76D03

Acknowledgments

The authors would like to thank Professor Lili Du for many helpful discussions and suggestions. The authors would also like to thank the referee for his/her pertinent comments and advice.

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China [Nos. 61563044, 71702119, 61762074], National Natural Science Foundation of Qinghai Province [No. 2017-ZJ-902], Open Research Fund Program of State Key Laboratory of Hydroscience and Engineering [No. sklhse-2017-A-05], The Science and Technology Major Project of Qinghai Province Natural Science Foundation [No. 2015-SF-A4-3].