## Global Regularity for the 2D Micropolar Rayleigh-Bénard Convection System with Velocity Zero Dissipation and Temperature Fractional Diffusion

Li Changhao,, Yuan Baoquan,*

School of Mathematics and Information Science, Henan Polytechnic University, Henan Jiaozuo 454000

Abstract

This paper studies the global regularity problem for the 2D micropolar Rayleigh-Bénard convection system with velocity zero dissipation, micro-rotation velocity Laplace dissipation and temperature fractional dissipation. By introducing two combined quantities and using the technique of Littlewood-Paley decomposition, this paper establishes the global regularity result of solutions to this system.

Keywords： Micropolar Rayleigh-Bénard; Global regularity; Besov space; Sobolev space

Li Changhao, Yuan Baoquan. Global Regularity for the 2D Micropolar Rayleigh-Bénard Convection System with Velocity Zero Dissipation and Temperature Fractional Diffusion[J]. Acta Mathematica Scientia, 2024, 44(4): 914-924

## 1 引言

$\left\{\begin{array}{l} \partial_{t} u+u \cdot \nabla u-(\mu+\chi) \Delta u+\nabla p=2 \chi \nabla \times \omega+e_{3} \theta \\ \partial_{t} \omega+u \cdot \nabla \omega-\nu \Delta \omega+4 \chi \omega-\eta \nabla \nabla \cdot \omega=2 \chi \nabla \times u \\ \partial_{t} \theta+u \cdot \nabla \theta-\kappa \Delta \theta=u \cdot e_{3} \\ \nabla \cdot u=0 \\ u(x, 0)=u_{0}(x), \omega(x, 0)=\omega_{0}(x), \theta(x, 0)=\theta_{0}(x) \end{array}\right.$

$\chi=0$, $\omega=0$, 则系统(1.1) 简化为 Bénard 方程. 当 Rayleigh-Bénard 对流项 $u\cdot e_{3}=0$ 时, Bénard 方程退化为Boussinesq方程. Boussinesq 方程可以用来模拟大气锋和海洋环流等地球物理流动, 并且对Rayleigh-Bénard 对流的研究具有重要意义. 由于其物理应用背景和数学上的重要意义, Boussinesq 方程近些年引起了广泛关注并取得了一些进展[2,8,11-14,27]. 当忽略温度的影响, 即 $\theta=0$ 时, 系统则系统(1.1) 简化为微极方程. 关于微极方程的更多研究进展可见文献[3,6,7,10,28]等.

$\left\{\begin{array}{l} \partial_{t} u+u \cdot \nabla u-(\mu+\chi) \Delta u+\nabla p=2 \chi \nabla \times \omega+e_{2} \theta, \\ \partial_{t} \omega+u \cdot \nabla \omega-\nu \Delta \omega+4 \chi \omega=2 \chi \nabla \times u, \\ \partial_{t} \theta+u \cdot \nabla \theta-\kappa \Delta \theta=u \cdot e_{2}, \\ \nabla \cdot u=0, \\ (u, \omega, \theta)(x, t)_{t=0}=\left(u_{0}, \omega_{0}, \theta_{0}\right)(x), \quad x \in \mathbb{R}^{2}.\end{array}\right.$

2020年, 徐夫义和迟美玲[26] 得到了温度零扩散 (即 $\kappa=0$) 时系统 (1.2) 的全局正则性. 2021 年, 王盛[24] 建立了没有速度耗散 (即 $\mu+\chi=0$) 时系统 (1.2) 的全局正则性. 随后, 邓利华和尚海锋[5]得到了只有速度耗散 (即 $\nu=\kappa=0$) 时系统 (1.2) 的全局正则性. 在本文中, 我们研究速度零耗散、温度为分数阶扩散的二维微极 Rayleigh-Bénard 对流系统的全局正则性, 具体方程如下所示

$\left\{\begin{array}{l} \partial_{t} u+u \cdot \nabla u+\nabla p=2 \chi \nabla \times \omega+e_{2} \theta, \\ \partial_{t} \omega+u \cdot \nabla \omega-\nu \Delta \omega+4 \chi \omega=2 \chi \nabla \times u, \\ \partial_{t} \theta+u \cdot \nabla \theta+\kappa \Lambda^{\gamma} \theta=u \cdot e_{2}, \\ \nabla \cdot u=0, \\ (u, \omega, \theta)(x, t)_{t=0}=\left(u_{0}, \omega_{0}, \theta_{0}\right)(x), \quad x \in \mathbb{R}^{2}.\end{array}\right.$

$h(x)=\mathcal{F}^{-1}\chi(x), h_{j}(x)=2^{dj}h(2^{j}x), \varphi_{j}(x)=2^{dj}\varphi(2^{j}x).$

$B_{p,q}^{s}=\{f(x)\in \mathcal{S}'(\mathbb{R}^{d});\|f\|_{B_{p,q}^{s}}<+\infty\},$

$\|f\|_{B_{p,q}^{s}}=\begin{cases}{ \begin{array}{ll} \bigg(\sum_{j\geq-1}2^{jsq}\|\Delta_{j} f\|_{L^{p}}^{q}\bigg)^{\frac{1}{q}},~&\mbox{若}~q<+\infty,\\[3mm] \sup_{j\geq-1}2^{js}\|\Delta_{j} f\|_{L^{p}},~&\mbox{若}~q=+\infty. \end{array} }\end{cases}$

(1) 如果对于整数 $j$ 和常数 $K>0$, $f$ 满足

${\rm supp}\hat{f}\subset\{\xi\in\mathbb{R}^{d}:~|\xi|\leq K2^{j}\},$

$\|(-\triangle)^{\alpha}f\|_{L^{q}}\leq C_{1}2^{2\alpha j+jd(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p}}.$

(2) 如果对于整数 $j$ 和常数 $0<K_{1}\leq K_{2}$, $f$ 满足

${\rm supp}\hat{f}\subset\{\xi\in\mathbb{R}^{d}:~K_{1}2^{j}\leq|\xi|\leq K_{2}2^{j}\},$

$C_{1}2^{2\alpha j}\|f\|_{L^{q}}\leq\|(-\triangle)^{\alpha}f\|_{L^{q}}\leq C_{2}2^{2\alpha j+jd(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p}}.$

$\frac{1}{r}=\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}.$

\begin{align*}&\|[\Lambda^{s},f]g\|_{L^{r}}\leq C(\|\nabla f\|_{L^{p_{1}}}\|\Lambda^{s-1}g\|_{L^{q_{1}}}+\|\Lambda^{s}f\|_{L^{p_{2}}}\|g\|_{L^{q_{2}}}),\\&\|\Lambda^{s}(f g)\|_{L^{r}}\leq C(\|\Lambda^{s}f\|_{L^{q_{1}}}\|g\|_{L^{p_{1}}}+\|f\|_{L^{q_{2}}}\|\Lambda^{s}g\|_{L^{p_{2}}}).\end{align*}

$\|\nabla^{s}{\rm e}^{\Delta t}f\|_{L^{q}}\leq Ct^{-\frac{s}{2}}t^{-\frac{d}{2}(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p}}.$

### 3.1 $\|\theta\|_{L^{\infty}}$ 的估计

$\|u\|^{2}_{L^{2}}+\|\omega\|^{2}_{L^{2}}+\|\theta\|^{2}_{L^{2}}+\int^{t}_{0}\|\nabla\omega(\cdot,\tau)\|^{2}_{L^{2}}\mathrm{d}\tau+\int^{t}_{0}\|\Lambda^{\frac{\gamma}{2}}\theta(\cdot,\tau)\|^{2}_{L^{2}}\mathrm{d}\tau\leq\|(u_{0},\omega_{0},\theta_{0})\|_{L^{2}}^{2}{\rm e}^{CT},$
$\|\theta\|_{L^{\infty}}\leq C{\rm e}^{CT}.$

$\|\nabla\omega\|_{L^{\infty}}+\int_{0}^{t}\|\nabla\theta\|_{L^{\infty}}\mathrm{d}\tau \leq C{\rm e}^{{\rm e}^{CT}}.$

$\omega={\rm e}^{t\Delta}\omega_{0}+\int_{0}^{t}{\rm e}^{(t-\tau)\Delta}(\Omega-2\omega-u\cdot\nabla\omega)\mathrm{d}\tau.$

$\begin{split}\|\nabla\omega\|_{L^{\infty}}&\leq \|\nabla {\rm e}^{t\Delta}\omega_{0}\|_{L^{\infty}}+\int_{0}^{t}\|\nabla {\rm e}^{(t-\tau)\Delta}(\Omega-2\omega-u\cdot\nabla\omega)\|_{L^{\infty}}\mathrm{d}\tau\\& \leq \|\nabla\omega_{0}\|_{L^{\infty}}+\int_{0}^{t}(t-\tau)^{-\frac{1}{2}}(\|\Omega\|_{L^{\infty}}+2\|\omega\|_{L^{\infty}}+\|u\|_{L^{\infty}}\|\nabla\omega\|_{L^{\infty}})\mathrm{d}\tau\\&\leq \|\nabla\omega_{0}\|_{L^{\infty}}+CT^{\frac{1}{3}}\bigg(\int_{0}^{t}({\rm e}^{{\rm e}^{CT}}+{\rm e}^{CT}\|\nabla\omega\|_{L^{\infty}})^{4}\mathrm{d}\tau\bigg)^{\frac{1}{4}},\end{split}$

$\|\nabla\omega\|_{L^{\infty}}^{4}\leq C\|\nabla\omega_{0}\|_{L^{\infty}}^{4}+CT^{\frac{4}{3}}\int_{0}^{t}({\rm e}^{{\rm e}^{CT}}+{\rm e}^{CT}\|\nabla\omega\|_{L^{\infty}}^{4})\mathrm{d}\tau.$

$\|\nabla\omega\|_{L^{\infty}}^{4}\mathrm{d}\tau\leq C{\rm e}^{{\rm e}^{CT}}.$

$\Lambda^{s}$ 作用于系统 (1.3), 再分别点乘 $(\Lambda^{s}u,\Lambda^{s}\omega,\Lambda^{s}\theta)$, 并在 $\mathbb{R}^{2}$ 上积分, 有

$\begin{matrix} & \ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(\|\Lambda^{s}u\|^{2}_{L^{2}}+\|\Lambda^{s}\omega\|^{2}_{L^{2}}+\|\Lambda^{s}\theta\|^{2}_{L^{2}}) +\|\Lambda^{s+1}\omega\|^{2}_{L^{2}}+2\|\Lambda^{s}\omega\|^{2}_{L^{2}}+\|\Lambda^{s+\frac{\gamma}{2}}\theta\|^{2}_{L^{2}} \\ &\leq-\int_{\mathbb{R}^{2}}[\Lambda^{s},u\cdot\nabla]u\cdot\Lambda^{s}u\mathrm{d}x +\int_{\mathbb{R}^{2}}\Lambda^{s}(\nabla\times\omega)\cdot\Lambda^{s}u\mathrm{d}x+\int_{\mathbb{R}^{2}}\Lambda^{s}(e_{2}\theta)\cdot\Lambda^{s}u\mathrm{d}x \\ & -\int_{\mathbb{R}^{2}}[\Lambda^{s},u\cdot\nabla]\omega\cdot\Lambda^{s}\omega \mathrm{d}x+\int_{\mathbb{R}^{2}}\Lambda^{s}(\nabla\times u)\cdot\Lambda^{s}\omega \mathrm{d}x \\ &~~~-\int_{\mathbb{R}^{2}}[\Lambda^{s},u\cdot\nabla]\theta\cdot\Lambda^{s}\theta \mathrm{d}x+\int_{\mathbb{R}^{2}}\Lambda^{s}(u\cdot e_{2})\cdot\Lambda^{s}\theta \mathrm{d}x \\ &=J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}+J_{7}. \end{matrix}$

$\begin{split} |J_{1}|&\leq C\|[\Lambda^{s},u\cdot\nabla]u\|_{L^{2}}\|\Lambda^{s}u\|_{L^{2}}\\ &\leq C(\|\nabla u\|_{L^{\infty}}\|\Lambda^{s}u\|_{L^{2}}+\|\Lambda^{s}u\|_{L^{2}}\|\nabla u\|_{L^{\infty}})\|\Lambda^{s}u\|_{L^{2}}\\ &\leq C\|\nabla u\|_{L^{\infty}}\|\Lambda^{s}u\|^{2}_{L^{2}}, \\ |J_{2}+J_{5}|&\leq C\|\Lambda^{s+1}\omega\|_{L^{2}}\|\Lambda^{s}u\|_{L^{2}}\leq \frac{1}{2}\|\Lambda^{s+1}\omega\|^{2}_{L^{2}}+C\|\Lambda^{s}u\|^{2}_{L^{2}}, \\ |J_{3}+J_{7}|&\leq C(\|\Lambda^{s}\theta\|^{2}_{L^{2}}+\|\Lambda^{s}u\|^{2}_{L^{2}}). \end{split}$

$\begin{split} |J_{4}|&\leq C(\|\nabla u\|_{L^{\infty}}\|\Lambda^{s}\omega\|_{L^{2}}+ \|\Lambda^{s}u\|_{L^{2}}\|\nabla\omega\|_{L^{\infty}})\|\Lambda^{s}\omega\|_{L^{2}}\\ &\leq C(\|\nabla u\|_{L^{\infty}}+\|\nabla \omega\|_{L^{\infty}})(\|\Lambda^{s}u\|_{L^{2}}^{2}+\|\Lambda^{s}\omega\|_{L^{2}}^{2}), \\ |J_{6}|&\leq C(\|\nabla u\|_{L^{\infty}}\|\Lambda^{s}\theta\|_{L^{2}}+ \|\Lambda^{s}u\|_{L^{2}}\|\nabla\theta\|_{L^{\infty}})\|\Lambda^{s}\theta\|_{L^{2}}\\ &\leq C(\|\nabla u\|_{L^{\infty}}+\|\nabla \theta\|_{L^{\infty}})(\|\Lambda^{s}u\|_{L^{2}}^{2}+\|\Lambda^{s}\theta\|_{L^{2}}^{2}). \end{split}$

$J_{1}-J_{7}$ 的估计代入 (3.20) 式, 利用对数型的 Sobolev 不等式 (参见引理 2.4), 我们有

$\begin{split} &\frac{\mathrm{d}}{\mathrm{d}t}(\|\Lambda^{s}u\|^{2}_{L^{2}}+\|\Lambda^{s}\omega\|^{2}_{L^{2}}+\|\Lambda^{s}\theta\|^{2}_{L^{2}}) +\|\Lambda^{s+1}\omega\|^{2}_{L^{2}}+\|\Lambda^{s+\frac{\gamma}{2}}\theta\|^{2}_{L^{2}}\\ \leq\ & C(\|\nabla u\|_{L^{\infty}}+\|\nabla\omega\|_{L^{\infty}}+\|\nabla\theta\|_{L^{\infty}}) (\|\Lambda^{s}u\|^{2}_{L^{2}}+\|\Lambda^{s}\omega\|^{2}_{L^{2}}+\|\Lambda^{s}\theta\|^{2}_{L^{2}})\\ \leq\ & C(1+\|\Omega\|_{L^{\infty}}\mathrm{log}(e+\|\Lambda^{s}u\|_{L^{2}})+ \|\nabla\omega\|_{L^{\infty}}+\|\nabla\theta\|_{L^{\infty}})\\ &\times(\|\Lambda^{s}u\|^{2}_{L^{2}}+\|\Lambda^{s}\omega\|^{2}_{L^{2}}+\|\Lambda^{s}\theta\|^{2}_{L^{2}}).\\ \end{split}$

$\begin{split} &(\|u\|^{2}_{H^{s}}+\|\omega\|^{2}_{H^{s}}+\|\theta\|^{2}_{H^{s}})+\int^{T}_{0}\|\nabla\omega\|^{2}_{H^{s}}\mathrm{d}\tau+\int^{T}_{0}\|\Lambda^{\frac{\gamma}{2}} \theta\|^{2}_{H^{s}}\mathrm{d}\tau\\ \leq\ & C(\|u_{0}\|_{H^{s}},\|\omega_{0}\|_{H^{s}},\|\theta_{0}\|_{H^{s}\cap B_{\infty,1}^{1-\gamma}},T). \end{split}$

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