Blow-up criterion for the zero-diffusive Boussinesq equations via the velocity components

• Main Author: Weihua Wang
• Format: Article
• Language: English
• Published: Texas State University 2015-03-01
• Series: Electronic Journal of Differential Equations
• Subjects: Zero-diffusive Boussinesq equations; blow up criterion; Lorentz spaces
• Online Access: http://ejde.math.txstate.edu/Volumes/2015/62/abstr.html

This article concerns the blow up for the smooth solutions of the three-dimensional Boussinesq equations with zero diffusivity. It is shown that if any two components of the velocity field $u$ satisfy $$ \int_0^T \frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} } {1+\ln ( e+\|\nabla u\|^2_{L^2}) } ds<\infty,\quad \frac{2}{q}+\frac{3}{p}=1,\quad 3<p<\infty, $$ then the local smooth solution $(u,\theta)$ can be continuously extended to $(0,T_1)$ for some $T_1>T$.