The 3D Boussinesq equations with regularity in the horizontal component of the velocity
Bull. Korean Math. Soc.
Published online December 17, 2019
Liu Qiao
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, P.R. of China
Abstract : This paper proves a new regularity criterion for solutions to the Cauchy problem of the 3D Boussinesq equations via one directional derivative of the horizontal component of the velocity field (i.e., $(\partial_{i}u_{1}; \partial_{j}u_{2}; 0)$ where $i, j\in\{1, 2, 3\}$) in the framework of the anisotropic Lebesgue spaces. More precisely, for $0<T<\infty$, if
\begin{align*}
&\int_{0}^{T}\!\!
\big( \left\|\left\| \partial_{i} u_{1}(t)\right\|_{L^{\alpha}_{x_{i}}}
\right\|_{L^{\beta}_{x_{\hat{i}}x_{\tilde{i}}}}^{\gamma}
\!+\!
\left\|\left\| \partial_{j} u_{2}(t)\right\|_{L^{\alpha}_{x_{j}}}
\right\|_{L^{\beta}_{x_{\hat{j}}x_{\tilde{j}}}}^{\gamma}
\big)
\text{d}t<\infty
\end{align*}
where $\frac{2}{\gamma}+\frac{1}{\alpha}+\frac{2}{\beta}=m\in[1,\frac{3}{2})$ and $\frac{3}{m}\leq \alpha\leq \beta < \frac{1}{m-1}$, then the corresponding solution $(u,\theta)$ to the 3D Boussinesq equations is regular on $[0,T]$. Here, $(i,\hat{i},\tilde{i})$ and $(j,\hat{j},\tilde{j})$ belong to the permutation group on the set $\mathbb{S}_{3}\!:=\{1,2,3\}$. This result reveals that the horizontal component of the
velocity field plays a dominant role in regularity theory of the Boussinesq equations.
Keywords : Boussinesq equations; regularity criterion; horizontal component; anisotropic Lebesgue spaces
MSC numbers : 35Q355, 35B65, 76D03
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