The 3D Boussinesq equations with regularity in the horizontal component of the velocity
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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.
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.
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