Blow-up criterion for the zero-diffusive Boussinesq equations via the velocity components
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 $$ \int_0^T \frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} } {1+\ln ( e+\|\nabla u\|^2_{L^2}) } d...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-03-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/62/abstr.html |
| Summary: | This article concerns the blow up for the smooth solutions of the
three-dimensional Boussinesq equations with zero diffusivity.
It is shown that if any two components of the velocity field $u$
satisfy
$$
\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 $T_1>T$. |
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| ISSN: | 1072-6691 |