Global regularity of 2D magnetic Bénard fluid equations with zero kinematic viscosity, almost Laplacian magnetic diffusion and thermal diffusivity

ABSTRACT

In this paper, we consider the global regularity of 2D magnetic Bénard fluid equations with almost magnetic diffusion and thermal diffusivity and without kinematic viscosity. We focus on this goal in two ways. In one way, the magnetic diffusion and thermal diffusivity are separately given by ${\mathfrak{D}}_2$ and ${\mathfrak{D}}_3$ two Fourier multipliers whose symbols $m_2$ and $m_3$ are respectively given by $m_2\left(\xi \right)\equiv |\xi {|}^2\log (e+|\xi {|}^2{)}^{\text{β}}$ and $m_3\left(\xi \right)\equiv |\xi {|}^2\log (e+|\xi {|}^2{)}^{\gamma }$. In another way, we generalize the previous case and the magnetic diffusion and thermal diffusivity are separately given by ${\mathfrak{L}}_2$ and ${\mathfrak{L}}_3$ two Fourier multipliers whose symbols $m_2$ and $m_3$ are respectively given by $m_2=|\xi {|}^2{g}_2\left(\xi \right)$ and $m_3=|\xi {|}^2{g}_3\left(\xi \right)$, where $g_j=g_j\left(|\xi |\right)$ $(j=2,3)$ are two radial non-decreasing smooth functions.

Keywords:

• Magnetic Bénard fluid equations
• global regularity
• global solution
• Fourier multiplier

2010 Mathematics Subject Classifications:

• Primary: 76W05
• Secondary: 35Q35
• Third:76B03
• Forth: 35B65
• Fifth: 76D03

Acknowledgments

The author would like to express sincere gratitude to Professor Lili Du for guidance, constant encouragement and providing an excellent research environment. The authors would also like to thank the referee for his/her pertinent comments and advice.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The research of L. Ma was supported by the National Natural Science Foundation of China (Grant Numbers 11571243, 11971331), China Scholarship Council (Grant Number 202008515084) and Teacher's development Scientific Research Staring Foundation of Chengdu University of Technology (Grant Number 10912-KYQD2019_07717).