On the improved blow-up criterion for the 2D zero diffusivity Boussinesq equations with temperature-dependent viscosity

Abstract

This paper examines the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity in the whole space. Whether or not its classical solutions global in time existence is a difficult problem and remains open. The main goal of this paper is to establish several blow-up criteria. More precisely, it is proved that is the maximal time interval if and only if the viscosity satisfies

or the vorticity satisfies

As a direct application of the above results, several other blow-up criteria in the Besov spaces and the multiplier spaces are also established.

Keywords:

• 2D Boussinesq equations
• variable viscosity
• blow-up criterion

AMS Subject Classifications:

• 35Q35
• 35B65
• 76D03

Acknowledgements

The author is grateful to the anonymous referees and the associated editor for their constructive comments and helpful suggestions that greatly improved the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.