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On the improved blow-up criterion for the 2D zero diffusivity Boussinesq equations with temperature-dependent viscosity. (English) Zbl 1398.35184
Summary: 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 \(0<T<\infty\) is the maximal time interval if and only if the viscosity \(\nu(\theta)\) satisfies \[ \int^T_0 \| \nabla\nu(\theta) (t)\|^2_{\mathrm{BMO}}\mathrm{d}t=\infty, \] or the vorticity \(\omega\) satisfies \[ \int^T_0 \|\omega (t)\|^2_{\mathrm{BMO}}\mathrm{d}t=\infty. \] As a direct application of the above results, several other blow-up criteria in the Besov spaces and the multiplier spaces are also established.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B44 Blow-up in context of PDEs
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