The 3D Boussinesq Equations with Regularity in One Directional Derivative of the Pressure

Abstract

This work establishes a new logarithmical improved regularity criterion for the 3D Boussinesq equations in terms of one directional derivative of the pressure (i.e., $\partial _{3}P$) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if

$$\begin{aligned}&\displaystyle \int _{0}^{T}\frac{\left\| \Vert \partial _{3}P(\cdot ,t)\Vert _{L^{\gamma }_{x_{3}}} \right\| _{L^{\alpha }_{x_{1}x_{2}}}^{\beta }}{1+\ln (1+\Vert \theta (\cdot ,t)\Vert _{L^{4}})}\text {d}t<\infty ,\\&\text {where } \frac{2}{\beta }+\frac{1}{\gamma }+\frac{2}{\alpha }=k\in [2,3) \text { and }\frac{3}{k}\le \gamma \le \alpha < \frac{1}{k-2}, \end{aligned}$$

for some $T>0$, then the corresponding solution $(u,\theta )$ to the 3D Boussinesq equations is regular on [0, T].

Appendix

This appendix provides the proof of (1.7). For the convenience of future references, we write it as a lemma.

Lemma C.1

Let $1\le \gamma ,\alpha ,\xi ,a,t\le \infty ,1<s\le \infty$, and $0\le \theta \le 1$. Then there exists a positive constant C such that¹

$$\begin{aligned} \left| \int _{\mathbb {R}^{3}} f g h\mathrm{d}x \right| \le&C\left\| \left\| \partial _{1}f\right\| _{L_{x_{1}}^{\gamma }}\right\| _{L^{\alpha }_{x_{2}x_{3}}}^{\frac{1}{s}} \left\| \left\| \partial _{1}f\right\| _{L_{x_{1}}^{\gamma }}\right\| _{L_{x_{2}x_{3}}^{\theta (s-1)t}}^{\frac{\theta (s-1)}{s}} \left\| \left\| f\right\| _{L_{x_{1}}^{\xi }}\right\| _{L_{x_{2}x_{3}}^{(1-\theta )(s-1)a}}^{\frac{(1-\theta )(s-1)}{s}}\nonumber \\&\times \Vert g\Vert _{L^{2}}^{\frac{s-1}{s}}\Vert (\partial _{2},\partial _{3})g\Vert _{L^{2}}^\frac{1}{s} \Vert h\Vert _{L^{2}}^{\frac{s-1}{s}}\Vert (\partial _{2},\partial _{3})h\Vert _{L^{2}}^\frac{1}{s}, \end{aligned}$$

(C.1)

for $f,g,h\in C_{0}^{\infty }(\mathbb {R}^{3})$, where $\gamma ,\alpha ,\xi ,s$ and $\theta$ satisfying

$$\begin{aligned}&\frac{1}{a}+\frac{1}{t}=\frac{\alpha -1}{\alpha }, \end{aligned}$$

(C.2)

and

$$\begin{aligned}&\frac{1}{(s-1)\gamma }+\frac{\theta }{\gamma }=\frac{1-\theta }{\xi (\gamma -1)}. \end{aligned}$$

(C.3)

Proof

The proof is essentially due to [7], and for the readers convenience, we give a simple proof. By using Hölder’s inequality, it follows that

$$\begin{aligned} \int _{\mathbb {R}}f g h\text {d}x_{1}\le \max _{x_{1}\in \mathbb {R}}|f|\left( \int _{\mathbb {R}}|g|^{2}\text {d}x_{1}\right) ^{\frac{1}{2}} \left( \int _{\mathbb {R}}|h|^{2}\text {d}x_{1}\right) ^{\frac{1}{2}}, \end{aligned}$$

and by using Hölder’s inequality again, one gets

$$\begin{aligned}&\left| \int _{\mathbb {R}^{3}} f g h \text {d}x\right| \le \left| \int _{\mathbb {R}^{2}} \left| \int _{\mathbb {R}}f g h\text {d}x_{1}\right| \text {d}x_{2}\text {d}x_{3}\right| \nonumber \\&\quad \le C\int _{\mathbb {R}^{2}}\left\{ \max _{x_{1}\in \mathbb {R}}|f|\left( \int _{\mathbb {R}}|g|^{2}\text {d}x_{1}\right) ^{\frac{1}{2}} \left( \int _{\mathbb {R}}|h|^{2}\text {d}x_{1}\right) ^{\frac{1}{2}}\right\} \text {d}x_{2}\text {d}x_{3}\nonumber \\&\quad \le C\left\{ \int _{\mathbb {R}^{2}}(\max _{x_{1}\in \mathbb {R}}|f|)^{s}\text {d}x_{2}\text {d}x_{3}\right\} ^{\frac{1}{s}} \left\{ \int _{\mathbb {R}^{2}}\left( \int _{\mathbb {R}}|g|^{2}\text {d}x_{1}\right) ^{\frac{s}{s-1}} \text {d}x_{2}\text {d}x_{3}\right\} ^{\frac{s-1}{2s}}\nonumber \\&\qquad \times \left\{ \int _{\mathbb {R}^{2}}\left( \int _{\mathbb {R}}|h|^{2}\text {d}x_{1}\right) ^{\frac{s}{s-1}} \text {d}x_{2}\text {d}x_{3}\right\} ^{\frac{s-1}{2s}}. \end{aligned}$$

(C.4)

By using Hölder’s and Gagliardo–Nirenberg inequalities, one obtains

$$\begin{aligned}&\left\{ \int _{\mathbb {R}^{2}}(\max _{x_{1}\in \mathbb {R}}|f|)^{s} \text {d}x_{2}\text {d}x_{3}\right\} ^{\frac{1}{s}} \le \left\{ \int _{\mathbb {R}^{3}}|f|^{s-1} |\partial _{1}f|\text {d}x_{1} \text {d}x_{2}\text {d}x_{3}\right\} ^{\frac{1}{s}}\nonumber \\&\quad \le C\left\| \Vert \partial _{1}f\Vert _{L_{x_{1}}^{\gamma }}\right\| _{L^{\alpha }_{x_{2}x_{3}}}^{\frac{1}{s}} \left\| \left\| f\right\| ^{s-1}_{L_{x_{1}}^{\frac{\gamma (s-1)}{\gamma -1}}} \right\| _{L_{x_{2},x_{3}}^{\frac{\alpha }{\alpha -1}}}^{\frac{1}{s}}\nonumber \\&\quad \le C\left\| \Vert \partial _{1}f\Vert _{L_{x_{1}}^{\gamma }}\right\| _{L^{\alpha }_{x_{2}x_{3}}}^{\frac{1}{s}} \left\| \left\| \partial _{1}f\right\| ^{\theta (s-1)}_{L_{x_{1}}^{\gamma }} \left\| f\right\| ^{(1-\theta )(s-1)}_{L_{x_{1}}^{\xi }} \right\| _{L_{x_{2},x_{3}}^{\frac{\alpha }{\alpha -1}}}^{\frac{1}{s}}\nonumber \\&\quad \le C \left\| \left\| \partial _{1}f\right\| _{L_{x_{1}}^{\gamma }} \right\| _{L^{\alpha }_{x_{2}x_{3}}}^{\frac{1}{s}} \left\| \left\| \partial _{1}f\right\| _{L_{x_{1}}^{\gamma }} \right\| _{L_{x_{2}x_{3}}^{\theta (s-1)t}}^{\frac{\theta (s-1)}{s}} \left\| \left\| f\right\| _{L_{x_{1}}^{\xi }} \right\| _{L_{x_{2}x_{3}}^{(1-\theta )(s-1)a}}^{\frac{(1-\theta )(s-1)}{s}} \end{aligned}$$

(C.5)

where we have used (C.2), (C.3) and the following inequality

$$\begin{aligned} |f(x_{1},x_{2},x_{3})|^{s}&\le C\int _{-\infty }^{x_{1}}|f(\tau ,x_{2},x_{3})|^{s-1} |\partial _{1}f(\tau ,x_{2},x_{3})|\text {d}\tau \\&\le C\int _{\mathbb {R}}|f(\tau ,x_{2},x_{3})|^{s-1} |\partial _{1}f(\tau ,x_{2},x_{3})|\text {d}\tau . \end{aligned}$$

Similarly, for $s>1$, one has

$$\begin{aligned} \left\{ \int _{\mathbb {R}^{2}}\left( \int _{\mathbb {R}}|g|^{2}\text {d}x_{1}\right) ^{\frac{s}{s-1}} \text {d}x_{2}\text {d}x_{3}\right\} ^{\frac{s-1}{2s}}&\le \left\{ \int _{\mathbb {R}}\left( \int _{\mathbb {R}^{2}}|g|^{\frac{2s}{s-1}} \text {d}x_{2}\text {d}x_{3}\right) ^{\frac{s-1}{s}}\text {d}x_{1}\right\} ^{\frac{1}{2}}\nonumber \\&\le C\Vert g\Vert _{L^{2}}^{\frac{s-1}{s}}\Vert (\partial _{2},\partial _{3})g\Vert _{L^{2}}^\frac{1}{s}; \end{aligned}$$

(C.6)

$$\begin{aligned} \left\{ \int _{\mathbb {R}^{2}}\left( \int _{\mathbb {R}}|h|^{2}\text {d}x_{1}\right) ^{\frac{s}{s-1}} \text {d}x_{2}\text {d}x_{3}\right\} ^{\frac{s-1}{2s}}&\le \left\{ \int _{\mathbb {R}}\left( \int _{\mathbb {R}^{2}}|h|^{\frac{2s}{s-1}} \text {d}x_{2}\text {d}x_{3}\right) ^{\frac{s-1}{s}}\text {d}x_{1}\right\} ^{\frac{1}{2}}\nonumber \\&\le C\Vert h\Vert _{L^{2}}^{\frac{s-1}{s}}\Vert (\partial _{2},\partial _{3})h\Vert _{L^{2}}^\frac{1}{s}. \end{aligned}$$

(C.7)

Inserting (C.5), (C.6) and (C.7) into (C.4), it follows that (C.1) holds. Thus we complete the proof of Lemma C.1. $\square$