## 分数阶扩散的MHD-Boussinesq系统的全局正则性

1 三峡大学理学院 湖北宜昌 443002

2 三峡大学数学研究中心 湖北宜昌 443002

## Global Regularity for the MHD-Boussinesq System with Fractional Diffusion

Yang Jing,1, Deng Xuemei,1,2, Zhou Yanping,1,2

1 College of Science, China Three Gorges University, Hubei Yichang 443002

2 Three Gorges Mathematical Research Center, China Three Gorges University, Hubei Yichang 443002

 基金资助: 国家自然科学基金.  11901346国家自然科学基金.  11871305三峡大学学位论文培优基金.  2021SSPY148

 Fund supported: the NSFC.  11901346the NSFC.  11871305the Research Fund for Excellent Dissertation of China Three Gorges University.  2021SSPY148

Abstract

In this paper, we investigate the $n$-dimensional $(n\geq2)$ Magnetohydrodynamics-Boussinesq system with fractional diffusion. When the nonnegative constants $\alpha, \beta$ and $\gamma$ satisfy $\alpha\geq\frac{1}{2}+\frac{n}{4}, \ \alpha+\beta\geq 1+\frac{n}{2}$ and $\alpha+\gamma\geq\frac{n}{2}$, by using the energy methods, we obtain the global existence and uniqueness of solution for the system, which generalizes the existing result.

Keywords： MHD-Boussinesq system ; Global regularity ; Fractional diffusion

Yang Jing, Deng Xuemei, Zhou Yanping. Global Regularity for the MHD-Boussinesq System with Fractional Diffusion. Acta Mathematica Scientia[J], 2021, 41(6): 1805-1815 doi:

## 1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} u_t+u\cdot\nabla u+\nabla p+\Lambda^{2\alpha}u = h\cdot\nabla h+\rho e_n, \\ h_t+u\cdot\nabla h-h\cdot\nabla u+\Lambda^{2\beta}h = 0, \\ \rho_t+u\cdot\nabla\rho+\Lambda^{2\gamma}\rho = 0, \\ {\rm div} u = {\rm div} h = 0, \\ ( u, h, \rho)(x, 0) = ( u_0, h_0, \rho_0), \end{array} \right. \end{eqnarray}$

## 2 主要结果

$$$\alpha\geq\frac{1}{2}+\frac{n}{4}, \ \alpha+\beta\geq 1+\frac{n}{2}, \ \alpha+\gamma\geq\frac{n}{2},$$$

$\begin{eqnarray} &&(u, h, \rho)\in L^{\infty}(0, T;H^s( {\Bbb R} ^ n )), \ u\in L^2(0, T;H^{s+\alpha}({\Bbb R} ^ n)), \\ && h\in L^2(0, T;H^{s+\beta}({\Bbb R} ^ n)), \ \rho\in L^2(0, T;H^{s+\gamma}({\Bbb R} ^ n)). \end{eqnarray}$

$\begin{eqnarray} &&\|\Lambda^s{u(t)}\|^2_{L^2} + \|\Lambda^s{h(t)}\|^2_{L^2} + \|\Lambda^s{\rho(t)}\|^2_{L^2} + \int_{0}^{t}(\|\Lambda^{\alpha+s}u(\tau)\|^2_{L^2} +\|\Lambda^{\beta+s}h(\tau)\|^2_{L^2} {}\\ &&+\|\Lambda^{\gamma+s}\rho(\tau)\|^2_{L^2}){\rm d}\tau \leq C. \end{eqnarray}$

证明将分为三个步骤: $H^0$估计, $H^1$估计及$H^s \ (s>1)$估计.

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|u\|^2_{L^2}+\|h\|^2_{L^2}+\|\rho\|^2_{L^2}) + \|\Lambda^{\alpha}u\|^2_{L^2}+\|\Lambda^{\beta}h\|^2_{L^2}+\|\Lambda^{\gamma}\rho\|^2_{L^2} \\ & = &\int_{{\Bbb R} ^ n}h\cdot\nabla h\cdot u \ {\rm d}x+\int_{{\Bbb R} ^ n}\rho e_n\cdot u \ {\rm d}x+\int_{{\Bbb R} ^ n} h\cdot\nabla u\cdot h \ {\rm d}x. \end{eqnarray}$

$\begin{eqnarray} &&\int_{{\Bbb R} ^ n}h\cdot\nabla h\cdot u\ {\rm d}x+\int_{{\Bbb R} ^ n} h\cdot\nabla u\cdot h\ {\rm d}x\\ & = &\int_{{\Bbb R} ^ n}h\cdot\nabla h\cdot u\ {\rm d}x-\int_{{\Bbb R} ^ n}{\rm div}(h\otimes h)\cdot u\ {\rm d}x \\ & = &\int_{{\Bbb R} ^ n}h\cdot\nabla h\cdot u \ {\rm d}x-\int_{{\Bbb R} ^ n}h\cdot\nabla h\cdot u \ {\rm d}x = 0. \end{eqnarray}$

$$$\int_{{\Bbb R} ^ n}\rho e_n\cdot u\ {\rm d}x \leq \|\rho\|_{L^2}\|u\|_{L^2}\leq\frac{1}{2}\|\rho\|^2_{L^2}+\frac{1}{2}\|u\|^2_{L^2}.$$$

$\begin{eqnarray} && \frac{1}{2}\frac{\rm d}{{\rm d}t}(\|u\|^2_{L^2}+\|h\|^2_{L^2}+\|\rho\|^2_{L^2})+(\|\Lambda^{\alpha}u\|^2_{L^2}+\|\Lambda^{\beta}h\|^2_{L^2} +\|\Lambda^{\gamma}\rho\|^2_{L^2}){}\\ &\leq&\frac{1}{2}(\|\rho\|^2_{L^2}+\|u\|^2_{L^2}). \end{eqnarray}$

$\begin{eqnarray} &&\|u(t)\|^2_{L^2}+\|h(t)\|^2_{L^2}+\|\rho(t)\|^2_{L^2}{}\\ &&+2\int_{0}^{t}(\|\Lambda^{\alpha}u(\tau)\|^2_{L^2} +\|\Lambda^{\beta}h(\tau)\|^2_{L^2}+\|\Lambda^{\gamma}\rho(\tau)\|^2_{L^2}) {\rm d}\tau\leq C. \end{eqnarray}$

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\nabla u\|^2_{L^2}+\|\nabla h\|^2_{L^2})+\|\Lambda^{\alpha+1}u\|^2_{L^2}+\|\Lambda^{\beta+1}h\|^2_{L^2}\\ & = &-\int_{{\Bbb R} ^ n}\partial_k u^i\partial_i u^j\partial_k u^j\ {\rm d}x-\int_{{\Bbb R} ^ n}(h\cdot\nabla h)\cdot\Delta u\ {\rm d}x-\int_{{\Bbb R} ^ n}\rho e_n\cdot\Delta u \ {\rm d}x\\ &&-\int_{{\Bbb R} ^ n}\partial_k u^i\partial_i h^j\partial_k h^j\ {\rm d}x-\int_{{\Bbb R} ^ n}(h\cdot\nabla u)\cdot\Delta h\ {\rm d}x\\ &: = &I_1+I_2+I_3+I_4+I_5. \end{eqnarray}$

$\begin{eqnarray} \|\nabla u\|_{L^3}\leq C\|\nabla u\|^a_{L^2}\|\Lambda^\alpha u\|^b_{L^2}\|\Lambda^{\alpha+1} u\|^c_{L^2}, \end{eqnarray}$

$\begin{eqnarray} I_1&\leq& \|\nabla u\|^3_{L^3}\leq C\|\nabla u\|^{3a}_{L^2}\|\Lambda^\alpha u\|^{3b}_{L^2}\|\Lambda^{\alpha+1}u\|^{3c}_{L^2}\\ &\leq&\varepsilon\|\Lambda^{\alpha+1}u\|^2_{L^2}+C\|\Lambda^\alpha u\|^{\frac{4\alpha}{6\alpha-2-n}}_{L^2}\|\nabla u\|^2_{L^2}. \end{eqnarray}$

$\beta<\frac{n}{2}$时, 注意到$\alpha+\beta\geq 1+\frac{n}{2}$, 有

$\begin{eqnarray} I_4&\leq& C\|\nabla h\|_{L^2}\|\nabla u\|_{L^{\frac{n}{\beta}}}\|\nabla h\|_{L^{\frac{2n}{n-2\beta}}}\\ &\leq& C\|\nabla h\|_{L^2}\|\Lambda^{\frac{n}{2}-\beta+1}u\|_{L^2}\|\Lambda^{\beta+1} h\|_{L^2}\\ &\leq&\varepsilon\|\Lambda^{\beta+1} h\|^2_{L^2}+C(\|u\|^2_{L^2}+\|\Lambda^{\alpha}u\|^2_{L^2})\|\nabla h\|^2_{L^2}. \end{eqnarray}$

$\beta>\frac{n}{2}$时, 有

$\begin{eqnarray} I_4&\leq &C\|\nabla h\|_{L^2}\|\nabla u\|_{L^2}\|\nabla h\|_{L^{\infty}}\\&\leq & C\|\nabla h\|_{L^2}(\|u\|_{L^2}+\|\Lambda^{\alpha} u\|_{L^2})(\|h\|_{L^2}+\|\Lambda^{\beta+1}h\|_{L^2})\\&\leq &\varepsilon\|\Lambda^{\beta+1}h\|^2_{L^2}+C\|h\|^2_{L^2}+C(\|u\|^2_{L^2}+\|\Lambda^{\alpha}u\|^2_{L^2})\|\nabla h\|^2_{L^2}. \end{eqnarray}$

$\begin{eqnarray} I_2+I_5& = &-\int_{{\Bbb R} ^ n}(h^i\partial_i h^j)\partial_k\partial_ku^j\ {\rm d}x- \int_{{\Bbb R} ^ n}(h^i\partial_i u^j)\partial_k\partial_k h^j\ {\rm d}x\\ & = &\int_{{\Bbb R} ^ n}\partial_k(h^i\partial_i h^j)\partial_k u^j\ {\rm d}x+\int_{{\Bbb R} ^ n} \partial_k(h^i\partial_i u^j)\partial_k h^j\ {\rm d}x\\ & = &\int_{{\Bbb R} ^ n}\partial_k h^i\partial_i h^j\partial_k u^j\ {\rm d}x+\int_{{\Bbb R} ^ n}\partial_k h^i\partial_i u^j\partial_k h^j\ {\rm d}x\\ &\leq &\varepsilon\|\Lambda^{\beta+1}h\|^2_{L^2}+C\|h\|^2_{L^2}+C(\|u\|^2_{L^2}+\|\Lambda^{\alpha}u\|^2_{L^2})\|\nabla h\|^2_{L^2}. \end{eqnarray}$

$\begin{eqnarray} I_3& = &-\int_{{\Bbb R} ^ n}\Lambda^{1-\alpha}(\rho e_n)\cdot\Lambda^{\alpha-1}\Delta u\ {\rm d}x\\ &\leq & C\|\Lambda^{1-\alpha}\rho\|_{L^2}\|\Lambda^{\alpha-1}\Delta u\|_{L^2} \\ {} & \leq& C(\|\rho\|_{L^2} +\|\Lambda^{\gamma}\rho\|_{L^2})\|\Lambda^{\alpha+1} u\|_{L^2}\\ &\leq &\varepsilon\|\Lambda^{\alpha+1} u\|^2_{L^2}+C(\|\rho\|^2_{L^2}+\|\Lambda^{\gamma}\rho\|^2_{L^2}). \end{eqnarray}$

$\begin{eqnarray} I_3&\leq &C\|\rho\|_{L^2}\|\Delta u\|_{L^2} \leq C\|\rho\|_{L^2}\|u\|^{\frac{\alpha-1}{\alpha+1}} _{L^2}\|\Lambda^{\alpha+1}u\|^{\frac{2}{\alpha+1}}_{L^2}\\ &\leq&\varepsilon\|\Lambda^{\alpha+1}u\|^2_{L^2}+C(\|\rho\|^2_{L^2}+\|u\|^2_{L^2}). \end{eqnarray}$

$\begin{eqnarray} \|\nabla u(t)\|^2_{L^2}+\|\nabla h(t)\|^2_{L^2}+\int_{0}^{t}(\|\Lambda^{\alpha+1}u(\tau)\|^2_{L^2}+\|\Lambda^{\beta+1}h(\tau)\|^2_{L^2}){\rm d}\tau\leq C. \end{eqnarray}$

$\begin{eqnarray} \frac{1}{2}\frac{\rm d}{{\rm d}t}\|\nabla\rho\|^2_{L^2}+\|\Lambda^{\gamma+1}\rho\|^2_{L^2} = -\int_{{\Bbb R} ^ n}\partial_k u^i\partial_i \rho^j\partial_k\rho^j\ {\rm d}x: = M_1. \end{eqnarray}$

$\begin{eqnarray} M_1&\leq &C\|\nabla\rho\|_{L^2}\|\nabla u\|_{L^{\frac{n}{\gamma}}}\|\nabla\rho\|_{L^{\frac{2n}{n-2\gamma}}}\\ &\leq& C\|\nabla\rho\|_{L^2}\|\Lambda^{\frac{n}{2}-\gamma+1}u\|_{L^2}\|\Lambda^{\gamma+1}\rho\|_{L^2}\\ &\leq&\varepsilon\|\Lambda^{\gamma+1}\rho\|^2_{L^2}+C(\|u\|^2_{L^2}+\|\Lambda^{\alpha+1}u\|^2_{L^2})\|\nabla\rho\|^2_{L^2}. \end{eqnarray}$

$\gamma>\frac{n}{2}$, 则有

$\begin{eqnarray} M_1&\leq& C\|\nabla\rho\|_{L^2}\|\nabla u\|_{L^2}\|\nabla\rho\|_{L^{\infty}}\\&\leq &C\|\nabla\rho\|_{L^2}(\|u\|_{L^2}+\|\Lambda^{\alpha+1}u\|_{L^2})(\|\rho\|_{L^2}+\|\Lambda^{\gamma+1} \rho\|_{L^2})\\ &\leq&\varepsilon\|\Lambda^{\gamma+1}\rho\|^2_{L^2}+C\|\rho\|^2_{L^2}+C(\|u\|^2_{L^2} +\|\Lambda^{\alpha+1}u\|^2_{L^2})\|\nabla\rho\|^2_{L^2}. \end{eqnarray}$

$\begin{eqnarray} \frac{\rm d}{{\rm d}t}\|\nabla\rho\|^2_{L^2}+\|\Lambda^{\gamma+1}\rho\|^2_{L^2}\leq C(\|u\|^2_{L^2}+\|\Lambda^{\alpha+1}u\|^2_{L^2})\|\nabla\rho\|^2_{L^2}+C\|\rho\|^2_{L^2}. \end{eqnarray}$

$\begin{eqnarray} \|\nabla\rho(t)\|^2_{L^2}+\int_{0}^{t}\|\Lambda^{\gamma+1}\rho(\tau)\|^2_{L^2}{\rm d}\tau\leq C. \end{eqnarray}$

$\begin{eqnarray} K_2+K_5&\leq&\varepsilon\|\Lambda^{s+\alpha} u\|^2_{L^2}+\varepsilon\|\Lambda^{s+\beta} h\|^2_{L^2} +C\|\Lambda^s u\|^2_{L^2}+C\|\nabla h\|^2_{L^2}\\ &&+C\big(\|u\|^2_{L^2}+\| h\|^2_{L^2}+\|\nabla u\|^2_{L^2}+\|\Lambda^{\alpha+1} u\|^2_{L^2}+\|\Lambda^{\beta+1} h\|^2_{L^2}\big)\|\Lambda^s h\|^2_{L^2}. \end{eqnarray}$

$\beta>\frac{n}{2}$, 有

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