2维带部分粘性Tropical Climate方程的整体适定性

2维带部分粘性Tropical Climate方程的整体适定性

王文娟^,, 薛明香^,

^{安徽大学数学科学学院合肥 230601}

Global Regularity of the 2D Tropical Climate Model with Partial Dissipation

Wang Wenjuan^,, Xue Mingxiang^,

^{School of Mathematical Sciences, Anhui University, Hefei 230601}

通讯作者: 王文娟, wangwenjuan@ahu.edu.cn

收稿日期: 2020-10-16

基金资助:

国家自然科学基金. 12001004

Received: 2020-10-16

Fund supported:

the NSFC. 12001004

作者简介 About authors

薛明香,E-mail:xmxathf@163.com , E-mail：xmxathf@163.com

Abstract

In this paper, we study the global existence and regularity of the 2D generalized tropical climate model, which has the standard Laplacian term Δv in the first baroclinic mode and partial dissipation in the barotropic mode and the temperature equation.

Keywords： Tropical Climate model ; Global regularity ; Partial dissipation

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本文引用格式

王文娟, 薛明香. 2维带部分粘性Tropical Climate方程的整体适定性. 数学物理学报[J], 2021, 41(6): 1734-1749 doi:

Wang Wenjuan, Xue Mingxiang. Global Regularity of the 2D Tropical Climate Model with Partial Dissipation. Acta Mathematica Scientia[J], 2021, 41(6): 1734-1749 doi:

1 引言

本文主要研究2维带部分粘性的Tropical Climate方程

$\begin{equation} \left\{\begin{array}{l} \partial_t u_1 +u\cdot\nabla u_1-\mu_{11}\partial_{xx}u_1-\mu_{12}\partial_{yy}u_1 +\partial_x p+\partial_x v_1^2+\partial_y(v_1v_2) = 0, \\ \partial_t u_2 +u\cdot\nabla u_2-\mu_{21}\partial_{xx}u_2-\mu_{22}\partial_{yy}u_2 +\partial_y p+\partial_x (v_1v_2)+\partial_y v_2^2 = 0, \\ \partial_x u_1+\partial_y u_2 = 0, \\ \partial_t v+u\cdot\nabla v-\Delta v+\nabla \theta+v\cdot\nabla u = 0, \\ \partial_t \theta+u\cdot\nabla \theta-\eta_1\partial_{xx}\theta-\eta_2\partial_{yy}\theta +\nabla\cdot v = 0, \end{array}\right. \end{equation}$

(1.1)

其中 $u(x, y, t) = (u_1(x, y, t), u_2(x, y, t)), \ v(x, y, t) = (v_1(x, y, t), v_2(x, y, t))$ 分别表示速度的正压模型和第一斜压模型, $\theta = \theta(x, y, t)$ 和 $p = p(x, y, t)$ 表示温度和压力, 系数 $\mu_{ij}$ 和 $\eta_i (i, j = 1, 2)$ 是非负实数.

Tropical Climate方程是由Frierson-Majda-Pauluis^[1]通过对完全无粘的Primitive方程中的第一斜压模型采用Galerkin截断推导出来的. 下面我们简单总结一下Tropical Climate方程现有的研究成果. Li和Titi^[2]通过引入伪斜压速度 $\omega = v-\nabla(-\Delta)^{-1}\theta$ , 并利用对数型Gronwall不等式, 证明了当正压模型和第一斜压模型具有标准的Laplace耗散项而温度场没有耗散项时2维Tropical Climate方程整体适定. Wan和Ma^[3-4]证明了2维Tropical Climate方程小初值强解整体适定, Ye和Zhu^[5-6]证明了2维Tropical Climate方程中正压模型具有与温度场相关的粘性时存在唯一的整体光滑解. 近年来, 数学家开始着手研究分数阶Tropical Climate方程, 其中耗散项分别为 $\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$ 和 $\Lambda^{2\gamma}\theta$ . Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当 $\alpha>0, \ \beta = \gamma = 1$ 或 $\mu = 0, \ \beta>1, \ \gamma = 1$ 时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当 $\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$ 时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量 $H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$ 并运用对数型Sobolev不等式证明了当 $\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$ 或 $\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$ 时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大 $L^q_tL^p_x$ 正则性，并结合Besov空间理论证明了当 $\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$ 或 $\alpha = 2, \ \beta = \gamma = 0$ 时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16].

受文章[11]启发, 本文致力于研究2维带部分粘性Tropical Climate方程的整体存在性和正则性, 有下面几种情形.

定理1.1 设 $(u_0, v_0, \theta_0)\in H^2({{\Bbb R}} ^2), \nabla \cdot u_0 = 0$ , 且 $\mu_{12} = \mu_{21} = \eta_2 = 1, \mu_{11} = \mu_{22} = \eta_1 = 0$ , 则方程(1.1)存在唯一的整体光滑解 $(u, v, \theta)$ 且对任意 $t>0$ ,

$(u, v, \theta)\in L^{\infty}(0, t;H^2({{\Bbb R}} ^2) ).$

注1.1 若 $\mu_{12} = \mu_{21} = \eta_2 = 1, \mu_{11} = \mu_{22} = \eta_1 = 0$ , 方程(1.1)可以写成

$\begin{equation} \left\{\begin{array}{l} \partial_t u_1 +u\cdot\nabla u_1-\partial_{yy}u_1 +\partial_x p+\partial_x v_1^2+\partial_y(v_1v_2) = 0, \\ \partial_t u_2 +u\cdot\nabla u_2-\partial_{xx}u_2 +\partial_y p+\partial_x (v_1v_2)+\partial_y v_2^2 = 0, \\ \partial_x u_1+\partial_y u_2 = 0, \\ \partial_t v+u\cdot\nabla v-\Delta v+\nabla \theta+v\cdot\nabla u = 0, \\ \partial_t \theta+u\cdot\nabla \theta-\partial_{yy}\theta +\nabla\cdot v = 0.\end{array}\right. \end{equation}$

(1.2)

注1.2 若把定理1.1中粘性系数条件换成 $\mu_{12} = \mu_{21} = \eta_1 = 1, \mu_{11} = \mu_{22} = \eta_2 = 0$ , 结论仍然成立. 证明方法与定理1.1类似, 这里我们省去.

定理1.2 设 $(u_0, v_0, \theta_0)\in H^2({{\Bbb R}} ^2), \nabla \cdot u_0 = 0$ , 且 $\mu_{11} = \mu_{21} = \eta_2 = 1, \mu_{12} = \mu_{22} = \eta_1 = 0$ , 则方程(1.1)存在唯一的整体光滑解 $(u, v, \theta)$ 且对任意 $t>0$ ,

$(u, v, \theta)\in L^{\infty}(0, t;H^2({{\Bbb R}} ^2) ).$

注1.3 若 $\mu_{11} = \mu_{21} = \eta_2 = 1, \mu_{12} = \mu_{22} = \eta_1 = 0$ , 方程(1.1)可以写成

$\begin{equation} \left\{\begin{array}{l} \partial_t u_1 +u\cdot\nabla u_1-\partial_{xx}u_1 +\partial_x p+\partial_x v_1^2+\partial_y(v_1v_2) = 0, \\ \partial_t u_2 +u\cdot\nabla u_2-\partial_{xx}u_2 +\partial_y p+\partial_x (v_1v_2)+\partial_y v_2^2 = 0, \\ \partial_x u_1+\partial_y u_2 = 0, \\ \partial_t v+u\cdot\nabla v-\Delta v+\nabla \theta+v\cdot\nabla u = 0, \\ \partial_t \theta+u\cdot\nabla \theta-\partial_{yy}\theta +\nabla\cdot v = 0.\end{array}\right. \end{equation}$

(1.3)

注1.4 令

$A = \left( \begin{array}{cc} \mu_{11}&\mu_{12}\\ \mu_{21}&\mu_{22} \end{array} \right), \ \ B = \left( \begin{array}{cc} \eta_1\\ \eta_2 \end{array} \right).$

设 $(u_0, v_0, \theta_0)\in H^2({{\Bbb R}} ^2), \nabla \cdot u_0 = 0$ , 当

$(A|B) = \left( \begin{array}{ccc} 0&1&1\\ 0&1&0 \end{array} \right), \ (A|B) = \left( \begin{array}{ccc} 1&1&1\\ 0&0&0 \end{array} \right) \ \mbox{或} \ \ (A|B) = \left( \begin{array}{ccc} 0&0&0\\ 1&1&1 \end{array} \right),$

运用定理1.2同样的方法, 可以证明方程(1.1)有唯一的整体光滑解.

局部光滑解的存在性和唯一性, 利用压缩映照原理即可得证, 这里我们省去. 我们主要验证定理1.1和定理1.2中解 $(u, v, \theta)$ 的先验估计, 除了反复应用分部积分和Young不等式, 还经常用到下面各向异性的Sobolev等式.

引理1.1^[12] 假设 $f, \ g, \ h, \ \partial_x g, \ \partial_y h\in L^2({{\Bbb R}} ^2)$ , 则

$\begin{equation} \int_{{{\Bbb R}} ^2}fgh{\rm d}x{\rm d}y\leq C\|f\|_{L^2}\ \|g\|_{L^2}^{\frac{1}{2}}\ \|\partial_x g\|_{L^2}^{\frac{1}{2}}\ \|h\|_{L^2}^{\frac{1}{2}}\ \|\partial_y h\|_{L^2}^{\frac{1}{2}}, \end{equation}$

(1.4)

其中 $C>0$ 为常数.

最后我们需要提一下, 当

$(A|B) = \left( \begin{array}{ccc} 1&0&1\\ 1&0&0 \end{array} \right), \ (A|B) = \left( \begin{array}{ccc} 0&1&0\\ 0&1&1 \end{array} \right)$

时, 方程(1.1)中粘性项分别是 $\partial_{xx}u_1, \partial_{xx}u_2, \partial_{xx}\theta$ 和 $\partial_{yy}u_1, \partial_{yy}u_2, \partial_{yy}\theta$ . 但是这两种情形无法使用与定理1.1和定理1.2类似的方法和技巧来证明, 或许问题并不是简单, 希望在不久的将来我们可以解决.

2 定理1.1的证明

首先, 用基本的能量办法验证解 $(u, v, \theta)$ 的 $L^2$ 估计.

在方程 $(1.2)_1, (1.2)_2, (1.2)_4, (1.2)_5$ 的两边分别用 $u_1, u_2, v, \theta$ 作 $L^2$ 内积, 并结合分部积分法, 有

$\begin{equation} \frac{1}{2}\frac{\rm d}{{\rm d}t}\|(u, v, \theta)\|_{L^2}^2 + \|\partial_y u_1\|_{L^2}^2 + \|\partial_x u_2\|_{L^2}^2 + \|\nabla v\|_{L^2}^2+ \|\partial_y \theta\|_{L^2}^2 = 0, \end{equation}$

(2.1)

这里我们利用了下面的等式

$\int_{{{\Bbb R}} ^2} (u\cdot \nabla u_1) u_1 {\rm d}x{\rm d}y = \int_{{{\Bbb R}} ^2} (u\cdot \nabla u_2)u_2 {\rm d}x{\rm d}y = \int_{{{\Bbb R}} ^2} (u\cdot \nabla)v\cdot v {\rm d}x{\rm d}y = \int_{{{\Bbb R}} ^2} (u\cdot \nabla\theta)\theta {\rm d}x{\rm d}y = 0,$

$\int_{{{\Bbb R}} ^2}\big(\partial_x v_1^2+\partial_y(v_1v_2)\big)u_1{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}\big(\partial_x (v_1v_2)+\partial_y v_2^2\big)u_2{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2} (v\cdot \nabla)u\cdot v{\rm d}x{\rm d}y = 0,$

$\int_{{{\Bbb R}} ^2} \nabla\theta\cdot v{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2} (\nabla\cdot v)\theta {\rm d}x{\rm d}y = 0.$

再在方程(2.1)两边关于时间从 $0$ 到 $t$ 积分, 有

$\begin{equation} \|(u, v, \theta)\|_{L^2}^2 (t) + 2\int_0^t (\|\partial_y u_1\|_{L^2}^2 + \|\partial_x u_2\|_{L^2}^2 + \|\nabla v\|_{L^2}^2+ \|\partial_y \theta\|_{L^2}^2){\rm d}\tau = \|(u_0, v_0, \theta_0)\|_{L^2}^2, \end{equation}$

(2.2)

对任意 $t>0$ 成立.

其次, 我们给出解 $(u, v, \theta)$ 的 $H^1$ 估计.

令涡量 $\omega = \nabla\times u = \partial_xu_2-\partial_yu_1$ , 结合方程 $(1.2)_1$ 与 $(1.2)_2$ , 得到

$\begin{equation} \partial_t \omega+u\cdot\nabla \omega-\partial_{xxx}u_2+\partial_{yyy}u_1+\partial_{xx}(v_1v_2) +\partial_{xy}v_2^2-\partial_{xy}v_1^2-\partial_{yy}(v_1v_2) = 0. \end{equation}$

(2.3)

接下来, 在方程 $(2.3), (1.2)_4, (1.2)_5$ 的两边分别用 $\omega, -\Delta v, -\Delta\theta$ 作 $L^2$ 内积, 有

$\begin{eqnarray*} \nonumber &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\|(\omega, \nabla v, \nabla\theta )\|_{L^2}^2+\frac{1}{2}\|\partial_x \omega\|_{L^2}^2 +\frac{1}{2}\|\partial_y \omega\|_{L^2}^2 + \|\Delta v\|_{L^2}^2 +\|\partial_y \nabla \theta\|_{L^2}^2\\ \nonumber &\leq& -\int_{{{\Bbb R}} ^2}\partial_{xx}(v_1v_2)\omega {\rm d}x{\rm d}y - \int_{{{\Bbb R}} ^2}\partial_{xy}v_2^2\omega {\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\partial_{xy}v_1^2\omega {\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\partial_{yy}(v_1v_2)\omega {\rm d}x{\rm d}y\\ \nonumber &&+\int_{{{\Bbb R}} ^2}u\cdot\nabla v\cdot\Delta v{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}v\cdot\nabla u\cdot\Delta v {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}u\cdot\nabla \theta\Delta \theta {\rm d}x{\rm d}y\\ \nonumber & = :& I_1+I_2+I_3+I_4+I_5+I_6+I_7, \end{eqnarray*}$

这里我们利用了

$\begin{eqnarray} &&\int_{{{\Bbb R}} ^2}(-\partial_{xxx}u_2+\partial_{yyy}u_1) \omega {\rm d}x{\rm d}y \\ & = & \int_{{{\Bbb R}} ^2}(-\partial_{xxx}u_2+\partial_{yyy}u_1) (\partial_xu_2-\partial_yu_1){\rm d}x{\rm d}y\\ & = &\|\partial_{xx} u_2\|_{L^2}^2+\|\partial_{yy} u_2\|_{L^2}^2+\|\partial_{xx} u_1\|_{L^2}^2 +\|\partial_{yy} u_1\|_{L^2}^2\\ &\geq& \frac{1}{2}\|\partial_x \omega\|_{L^2}^2+\frac{1}{2}\|\partial_y \omega\|_{L^2}^2 \end{eqnarray}$

和

$\int_{{{\Bbb R}} ^2}\nabla\theta\cdot\Delta v{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}(\nabla\cdot v)\Delta \theta {\rm d}x{\rm d}y = 0.$

下面我们使用引理1.1中各向异性的Sobolev不等式和Young不等式, 分别估计 $I_1, \ I_2, \ I_3,$ $I_4, \ I_5, \ I_6$ 和 $I_7$ .

$\begin{eqnarray*} \nonumber I_1& = &-\int_{{{\Bbb R}} ^2}v_1\partial_{xx}v_2 \omega {\rm d}x{\rm d}y - 2\int_{{{\Bbb R}} ^2}\partial_x v_1\partial_x v_2 \omega {\rm d}x{\rm d}y - \int_{{{\Bbb R}} ^2}\partial_{xx}v_1v_2\omega {\rm d}x{\rm d}y\\ \nonumber & = :& I_{11}+ I_{12}+ I_{13}, \end{eqnarray*}$

其中

$\begin{eqnarray*} \nonumber I_{11}&\leq&C\|\partial_{xx}v_2\|_{L^2}\|v_1\|_{L^2}^{\frac{1}{2}} \|\partial_yv_1\|_{L^2}^{\frac{1}{2}} \|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\Delta v\|_{L^2}\|\nabla v\|_{L^2}^{\frac{1}{2}} \|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{36}\|\Delta v\|_{L^2}^2+\frac{1}{24}\|\partial_x \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2, \\ I_{12}&\leq&C\|\omega\|_{L^2}\|\partial_x v_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xx}v_1\|_{L^2}^{\frac{1}{2}} \|\partial_x v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} v_2\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\omega\|_{L^2}\|\nabla v\|_{L^2}\|\Delta v\|_{L^2}\\ \nonumber &\leq&\frac{1}{36}\|\Delta v\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2 \end{eqnarray*}$

和

$\begin{eqnarray*} \nonumber I_{13}&\leq&C\|\partial_{xx}v_1\|_{L^2}\|v_2\|_{L^2}^{\frac{1}{2}} \|\partial_yv_2\|_{L^2}^{\frac{1}{2}} \|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\Delta v\|_{L^2}\|\nabla v\|_{L^2}^{\frac{1}{2}} \|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{36}\|\Delta v\|_{L^2}^2+\frac{1}{24}\|\partial_x \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2, \end{eqnarray*}$

从而

$\begin{equation} I_1\leq \frac{1}{12}\|\Delta v\|_{L^2}^2+\frac{1}{12}\|\partial_x \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2. \end{equation}$

(2.4)

对于 $I_2$ , 有

$\begin{eqnarray*} \nonumber I_2& = &-2\int_{{{\Bbb R}} ^2}v_2\partial_{xy}v_2 \omega {\rm d}x{\rm d}y - 2\int_{{{\Bbb R}} ^2}\partial_x v_2\partial_y v_2 \omega {\rm d}x{\rm d}y\\ \nonumber & = :& I_{21}+ I_{22}, \\ \nonumber I_{21}&\leq&C\|\partial_{xy}v_2\|_{L^2}\|v_2\|_{L^2}^{\frac{1}{2}} \|\partial_yv_2\|_{L^2}^{\frac{1}{2}} \|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\Delta v\|_{L^2}\|\nabla v\|_{L^2}^{\frac{1}{2}} \|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{24}\|\Delta v\|_{L^2}^2+\frac{1}{12}\|\partial_x \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2, \\ \nonumber I_{22}&\leq&C\|\omega\|_{L^2}\|\partial_x v_2\|_{L^2}^{\frac{1}{2}} \|\partial_{xx}v_2\|_{L^2}^{\frac{1}{2}} \|\partial_y v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{yy} v_2\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\omega\|_{L^2}\|\nabla v\|_{L^2}\|\Delta v\|_{L^2}\\ \nonumber &\leq&\frac{1}{24}\|\Delta v\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2, \end{eqnarray*}$

故

$\begin{equation} I_2\leq \frac{1}{12}\|\Delta v\|_{L^2}^2+\frac{1}{12}\|\partial_x \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2. \end{equation}$

(2.5)

类似地

$\begin{equation} I_3\leq \frac{1}{12}\|\Delta v\|_{L^2}^2+\frac{1}{12}\|\partial_y \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2, \end{equation}$

(2.6)

$\begin{equation} I_4\leq \frac{1}{12}\|\Delta v\|_{L^2}^2+\frac{1}{12}\|\partial_y \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\| \omega\|_{L^2}^2. \end{equation}$

(2.7)

对于 $I_5$ , 有

$\begin{eqnarray} I_5& = &\int_{{{\Bbb R}} ^2}(u_1\partial_x v_1+u_2\partial_y v_1)\Delta v_1 {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}(u_1\partial_x v_2+u_2\partial_y v_2)\Delta v_2{\rm d}x{\rm d}y\\ & = :& I_{51}+ I_{52}, \end{eqnarray}$

又

$\begin{eqnarray} I_{51}&\leq&C\|\Delta v_1\|_{L^2}\|u_1\|_{L^2}^{\frac{1}{2}} \|\partial_yu_1\|_{L^2}^{\frac{1}{2}} \|\partial_x v_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xx} v_1\|_{L^2}^{\frac{1}{2}}\\ && +C\|\Delta v_1\|_{L^2}\|u_2\|_{L^2}^{\frac{1}{2}} \|\partial_xu_2\|_{L^2}^{\frac{1}{2}} \|\partial_y v_1\|_{L^2}^{\frac{1}{2}}\|\partial_{yy} v_1\|_{L^2}^{\frac{1}{2}}\\ &\leq& C\|\Delta v\|_{L^2}\Big(\|\partial_yu_1\|_{L^2}^{\frac{1}{2}} \|\nabla v\|_{L^2}^{\frac{1}{2}}\|\Delta v\|_{L^2}^{\frac{1}{2}}+ \|\partial_xu_2\|_{L^2}^{\frac{1}{2}} \|\nabla v\|_{L^2}^{\frac{1}{2}}\|\Delta v\|_{L^2}^{\frac{1}{2}}\Big)\\ &\leq& C\|\Delta v\|_{L^2}^{\frac{3}{2}}\|\nabla v\|_{L^2}^{\frac{1}{2}} \Big(\|\partial_yu_1\|_{L^2}^{\frac{1}{2}}+\|\partial_xu_2\| _{L^2}^{\frac{1}{2}}\Big)\\ &\leq&\frac{1}{24}\|\Delta v\|_{L^2}^2+ C\big(\|\partial_yu_1\|_{L^2}^2+\|\partial_xu_2\|_{L^2}^2\big) \|\nabla v\|_{L^2}^2, \end{eqnarray}$

同样地

$I_{52}\leq \frac{1}{24}\|\Delta v\|_{L^2}^2+ C\big(\|\partial_yu_1\|_{L^2}^2+\|\partial_xu_2\|_{L^2}^2\big)\|\nabla v\|_{L^2}^2,$

所以

$\begin{equation} I_5\leq \frac{1}{12}\|\Delta v\|_{L^2}^2+ C(\|\partial_yu_1\|_{L^2}^2+\|\partial_xu_2\|_{L^2}^2)\|\nabla v\|_{L^2}^2. \end{equation}$

(2.8)

考虑 $I_6$

$\begin{eqnarray*} \nonumber I_6& = &\int_{{{\Bbb R}} ^2}(v_1\partial_x u_1+v_2\partial_y u_1)\Delta v_1{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}(v_1\partial_x u_2+v_2\partial_y u_2)\Delta v_2 {\rm d}x{\rm d}y\\ \nonumber & = :& I_{61}+ I_{62}, \end{eqnarray*}$

因为

$\begin{eqnarray*} \nonumber I_{61}&\leq&C\|\Delta v_1\|_{L^2}\|v_1\|_{L^2}^{\frac{1}{2}} \|\partial_yv_1\|_{L^2}^{\frac{1}{2}} \|\partial_x u_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xx} u_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber && +C\|\Delta v_1\|_{L^2}\|v_2\|_{L^2}^{\frac{1}{2}} \|\partial_yv_2\|_{L^2}^{\frac{1}{2}} \|\partial_y u_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} u_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\Delta v\|_{L^2}\Big( \|\nabla v\|_{L^2}^{\frac{1}{2}}\|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_y \omega\|_{L^2}^{\frac{1}{2}}+ \|\nabla v\|_{L^2}^{\frac{1}{2}}\|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}}\Big)\\ \nonumber &\leq&\frac{1}{24}\|\Delta v\|_{L^2}^2+ \frac{1}{24}\|\partial_x \omega\|_{L^2}^2 + \frac{1}{24}\|\partial_y \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\|\omega\|_{L^2}^2, \\ I_{62}&\leq &\frac{1}{24}\|\Delta v\|_{L^2}^2+ \frac{1}{24}\|\partial_x \omega\|_{L^2}^2 + \frac{1}{24}\|\partial_y \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\|\omega\|_{L^2}^2, \end{eqnarray*}$

故

$\begin{equation} I_6\leq \frac{1}{12}\|\Delta v\|_{L^2}^2+ \frac{1}{12}\|\partial_x \omega\|_{L^2}^2 + \frac{1}{12}\|\partial_y \omega\|_{L^2}^2+ C\|\nabla v\|_{L^2}^2\|\omega\|_{L^2}^2. \end{equation}$

(2.9)

利用部分积分法, 有

$\begin{eqnarray} I_7& = &-\int_{{{\Bbb R}} ^2}\nabla\theta\cdot\nabla u\cdot\nabla\theta {\rm d}x{\rm d}y\\ & = &-\int_{{{\Bbb R}} ^2}\partial_x\theta\partial_x u_1\partial_x\theta {\rm d}x{\rm d}y-\int_{{{\Bbb R}} ^2}(\partial_x u_2+\partial_y u_1)\partial_x\theta\partial_y\theta {\rm d}x{\rm d}y-\int_{{{\Bbb R}} ^2}\partial_y\theta\partial_y u_2\partial_y\theta {\rm d}x{\rm d}y\\ & = :& I_{71}+ I_{72}+I_{73}, \end{eqnarray}$

同时

$\begin{eqnarray*} \nonumber I_{71}& = &\int_{{{\Bbb R}} ^2}\partial_x\theta\partial_y u_2\partial_x\theta {\rm d}x{\rm d}y = -2\int_{{{\Bbb R}} ^2}u_2\partial_x\theta\partial_{xy}\theta {\rm d}x{\rm d}y\\ \nonumber&\leq&C\|\partial_{xy}\theta\|_{L^2}\|u_2\| _{L^2}^{\frac{1}{2}}\|\partial_x u_2\|_{L^2}^{\frac{1}{2}} \|\partial_x \theta\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq& C\|\partial_y\nabla\theta\|_{L^2}^{\frac{3}{2}} \|\partial_xu_2\|_{L^2}^{\frac{1}{2}} \|\nabla \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{8}\|\partial_y\nabla\theta\|_{L^2}^2+ C\|\partial_x u_2\|_{L^2}^2\|\nabla \theta\|_{L^2}^2, \\ \nonumber I_{72}&\leq&C(\|\partial_x u_2\|_{L^2}+\|\partial_y u_1\|_{L^2}) \|\partial_x\theta\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}\theta\|_{L^2}^{\frac{1}{2}} \|\partial_y\theta\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}\theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C(\|\partial_x u_2\|_{L^2}+\|\partial_y u_1\|_{L^2})\|\nabla\theta\|_{L^2} \|\partial_y\nabla\theta\|_{L^2}\\ \nonumber &\leq&\frac{1}{4}\|\partial_y\nabla\theta\|_{L^2}^2+ C\left(\|\partial_x u_2\|_{L^2}^2+\|\partial_y u_1\|_{L^2}^2\right)\|\nabla\theta\|_{L^2}^2, \\ \nonumber I_{73}& = &2\int_{{{\Bbb R}} ^2}u_2\partial_y\theta\partial_{yy}\theta {\rm d}x{\rm d}y\\ \nonumber&\leq&C\|\partial_{yy}\theta\|_{L^2} \|u_2\|_{L^2}^{\frac{1}{2}}\|\partial_x u_2\|_{L^2}^{\frac{1}{2}} \|\partial_y \theta\|_{L^2}^{\frac{1}{2}}\|\partial_{yy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq&C\|\partial_y\nabla\theta\|_{L^2}^{\frac{3}{2}} \|\partial_x u_2\|_{L^2}^{\frac{1}{2}} \|\nabla \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{8}\|\partial_y\nabla\theta\|_{L^2}^2+ C\|\partial_x u_2\|_{L^2}^2\|\nabla \theta\|_{L^2}^2, \end{eqnarray*}$

从而

$\begin{equation} I_7\leq \frac{1}{2}\|\partial_y\nabla\theta\|_{L^2}^2+ C(\|\partial_x u_2\|_{L^2}^2+\|\partial_y u_1\|_{L^2}^2)\|\nabla\theta\|_{L^2}^2. \end{equation}$

(2.10)

综合估计式(2.4)–(2.10), 可得

$\begin{eqnarray} &&\frac{\rm d}{{\rm d}t}\|(\omega, \nabla v, \nabla\theta )\| _{L^2}^2+\frac{1}{2}\|\partial_x \omega\|_{L^2}^2 +\frac{1}{2}\|\partial_y \omega\|_{L^2}^2 + \|\Delta v\|_{L^2}^2 +\|\partial_y \nabla \theta\|_{L^2}^2\\ &\leq&C(\|\nabla v\|_{L^2}^2+\|\partial_x u_2\|_{L^2}^2+\|\partial_y u_1\|_{L^2}^2)\|(\omega, \nabla v, \nabla\theta )\|_{L^2}^2. \end{eqnarray}$

(2.11)

应用Gronwall不等式和(2.2)式, 可得

$\begin{eqnarray} &&\|(\omega, \nabla v, \nabla\theta)\|_{L^2}^2 (t) + \int_0^t (\|\partial_x \omega\|_{L^2}^2 + \|\partial_y \omega\|_{L^2}^2 + \|\Delta v\|_{L^2}^2+ \|\partial_y \nabla\theta\|_{L^2}^2)\ {\rm d}\tau \\ &\leq& C(t, \|(u_0, v_0, \theta_0)\|_{H^1}), \end{eqnarray}$

(2.12)

对任意 $t>0$ 成立.

最后我们证明 $\|(u, v, \theta)\|_{H^2}$ 有界.

用算子 $\nabla$ 作用于方程 $(2.3), (1.2)_4, (1.2)_5$ 的两边, 接着分别用 $\nabla \omega, -\nabla\Delta v, -\nabla\Delta \theta$ 作 $L^2$ 内积, 有

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\|(\nabla \omega, \Delta v, \Delta\theta )\|_{L^2}^2+\|\partial_{xx} \omega\|_{L^2}^2 +\|\partial_{yy} \omega\|_{L^2}^2 + \|\nabla\Delta v\|_{L^2}^2 +\|\partial_y \Delta \theta\|_{L^2}^2\\ & = & \int_{{{\Bbb R}} ^2}\partial_{xx}(v_1v_2)\Delta \omega {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}\partial_{xy}v_2^2 \Delta \omega {\rm d}x{\rm d}y-\int_{{{\Bbb R}} ^2}\partial_{xy}v_1^2 \Delta \omega {\rm d}x{\rm d}y\\ &&-\int_{{{\Bbb R}} ^2}\partial_{yy}(v_1v_2) \Delta \omega {\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}u\cdot\nabla \omega\Delta \omega {\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\nabla (u\cdot\nabla) v\nabla\Delta v{\rm d}x{\rm d}y\\ &&+ \int_{{{\Bbb R}} ^2}\nabla(v\cdot\nabla) u:\nabla\Delta v{\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}\nabla (u\cdot\nabla) v\cdot\nabla\Delta \theta {\rm d}x{\rm d}y\\ & = :& J_1+J_2+J_3+J_4+J_5+J_6+J_7+J_8, \end{eqnarray}$

这里我们使用了下面的等式

$\int_{{{\Bbb R}} ^2}\nabla(-\partial_{xxx}u_2+\partial_{yyy}u_1)\cdot\nabla \omega {\rm d}x{\rm d}y = \|\partial_{xx}\omega\|_{L^2}^2 +\|\partial_{yy}\omega\|_{L^2}^2$

和

$\int_{{{\Bbb R}} ^2}\nabla\nabla\theta:\nabla\Delta v{\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\nabla(\nabla\cdot v)\cdot\nabla\Delta\theta {\rm d}x{\rm d}y = 0.$

接下来我们利用部分积分法、各向异性的Sobolev不等式和Young不等式分别估计 $J_1, J_2, J_3, J_4, J_5, J_6, J_7$ 和 $J_8$ .

$\begin{eqnarray} J_1& = &\int_{{{\Bbb R}} ^2}v_1\partial_{xx}v_2 \Delta \omega {\rm d}x{\rm d}y + 2\int_{{{\Bbb R}} ^2}\partial_x v_1\partial_x v_2 \Delta \omega {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}\partial_{xx}v_1v_2 \Delta \omega {\rm d}x{\rm d}y\\ & = :& J_{11}+ J_{12}+ J_{13}, \end{eqnarray}$

其中

$\begin{eqnarray*} \nonumber J_{11}&\leq&C\|\Delta \omega\|_{L^2} \|v_1\|_{L^2}^{\frac{1}{2}} \|\partial_x v_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xx}v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xxy} v_2\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq& C\|\Delta\omega\|_{L^2}\|\Delta v\| _{L^2}^{\frac{1}{2}}\|\nabla\Delta v\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{84}\|\Delta\omega\|_{L^2}^2+ C\|\Delta v\|_{L^2}\|\nabla\Delta v\|_{L^2}\\ \nonumber&\leq&\frac{1}{42}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{42}\|\partial_{yy} \omega\|_{L^2}^2+ \frac{1}{24}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2, \\ J_{12}&\leq&C\|\Delta \omega\|_{L^2}\|\partial_x v_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xx}v_1\|_{L^2}^{\frac{1}{2}} \|\partial_x v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} v_2\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq& C\|\Delta \omega\|_{L^2}\|\Delta v\|_{L^2}\\ \nonumber&\leq&\frac{1}{42}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{42}\|\partial_{yy} \omega\|_{L^2}^2 +C\|\Delta v\|_{L^2}^2, \end{eqnarray*}$

类似地

$J_{13}\leq \frac{1}{42}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{42}\|\partial_{yy} \omega\|_{L^2}^2+ \frac{1}{24}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2,$

于是

$\begin{equation} J_1\leq \frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{14}\|\partial_{yy} \omega\|_{L^2}^2+ \frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2. \end{equation}$

(2.13)

对于 $J_2$ , 有

$\begin{eqnarray} J_2& = &2\int_{{{\Bbb R}} ^2}v_2\partial_{xy}v_2 \Delta \omega {\rm d}x{\rm d}y + 2\int_{{{\Bbb R}} ^2}\partial_x v_2\partial_y v_2 \Delta \omega {\rm d}x{\rm d}y\\ & = :& J_{21}+ J_{22}, \end{eqnarray}$

同时

$\begin{eqnarray*} \nonumber J_{21}&\leq&C\|\Delta \omega\|_{L^2} \|v_2\|_{L^2}^{\frac{1}{2}}\|\partial_xv_2\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xyy}v_2\| _{L^2}^{\frac{1}{2}}\\ \nonumber&\leq& C\|\Delta \omega\|_{L^2} \|\Delta v\|_{L^2}^{\frac{1}{2}} \|\nabla\Delta v\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{28}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{28}\|\partial_{yy} \omega\|_{L^2}^2+ \frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2, \\ J_{22}&\leq&C\|\Delta \omega\|_{L^2}\|\partial_x v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xx}v_2\|_{L^2}^{\frac{1}{2}} \|\partial_y v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{yy} v_2\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq&C\|\Delta \omega\|_{L^2}\|\Delta v\|_{L^2}\\ \nonumber&\leq&\frac{1}{28}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{28}\|\partial_{yy} \omega\|_{L^2}^2+C\|\Delta v\|_{L^2}^2, \end{eqnarray*}$

从而

$\begin{equation} J_2\leq \frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{14}\|\partial_{yy} \omega\|_{L^2}^2+ \frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2. \end{equation}$

(2.14)

同样地

$\begin{equation} J_3\leq \frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{14}\|\partial_{yy} \omega\|_{L^2}^2+ \frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2, \end{equation}$

(2.15)

$\begin{equation} J_4\leq \frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{14}\|\partial_{yy} \omega\|_{L^2}^2+ \frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2. \end{equation}$

(2.16)

下面估计 $J_5$ .

$\begin{eqnarray} J_5& = &\int_{{{\Bbb R}} ^2}u_1\partial_x \omega\Delta \omega {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}u_2\partial_y \omega \Delta \omega {\rm d}x{\rm d}y\\ &\leq&C\|\Delta \omega\|_{L^2}\|u_1\|_{L^2}^{\frac{1}{2}} \|\partial_y u_1\|_{L^2}^{\frac{1}{2}} \|\partial_x \omega\|_{L^2}^{\frac{1}{2}} \|\partial_{xx}\omega\|_{L^2}^{\frac{1}{2}}\\ &&+C\|\Delta \omega\|_{L^2}\|u_2\|_{L^2}^{\frac{1}{2}} \|\partial_x u_2\|_{L^2}^{\frac{1}{2}}\|\partial_y \omega\|_{L^2}^{\frac{1}{2}}\|\partial_{yy}\omega\| _{L^2}^{\frac{1}{2}}\\ &\leq&C\|\Delta \omega\|_{L^2}^{\frac{3}{2}}\|\nabla \omega\|_{L^2}^{\frac{1}{2}}\\ &\leq&\frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{14}\|\partial_{yy} \omega\|_{L^2}^2+C\|\nabla \omega\|_{L^2}^2. \end{eqnarray}$

(2.17)

对于 $J_6$ , 有

$\begin{eqnarray} J_6& = &\int_{{{\Bbb R}} ^2}\partial_x(u_1\partial_x v_1+u_2\partial_y v_1)\partial_x\Delta v_1 {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}\partial_x(u_1\partial_x v_2+u_2\partial_y v_2)\partial_x\Delta v_2 {\rm d}x{\rm d}y\\ &&+\int_{{{\Bbb R}} ^2}\partial_y(u_1\partial_x v_1+u_2\partial_y v_1)\partial_y\Delta v_1{\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}\partial_y(u_1\partial_x v_2+u_2\partial_y v_2)\partial_y\Delta v_2 {\rm d}x{\rm d}y\\ & = :& J_{61}+ J_{62}+ J_{63}+J_{64}, \end{eqnarray}$

其中

$\begin{eqnarray} J_{61}& = &\int_{{{\Bbb R}} ^2}(u_1\partial_{xx}v_1+u_2\partial_{xy}v_1) \partial_x\Delta v_1{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}(\partial_x u_1\partial_xv_1+\partial_x u_2\partial_y v_1)\partial_x\Delta v_1{\rm d}x{\rm d}y\\ & = :& J_{611}+ J_{612}, \end{eqnarray}$

并且

$\begin{eqnarray*} \nonumber J_{611}&\leq&C\|\partial_x\Delta v_1\|_{L^2}\|u_1\|_{L^2}^{\frac{1}{2}} \|\partial_y u_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xx} v_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xxx}v_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_x\Delta v_1\|_{L^2}\|u_2\|_{L^2}^{\frac{1}{2}} \|\partial_x u_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} v_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xyy}v_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq& C\|\nabla\Delta v\|_{L^2}^{\frac{3}{2}}\|\Delta v\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{96}\|\nabla\Delta v\|_{L^2}^2+C\|\Delta v\|_{L^2}^2, \\ J_{612}&\leq&C\|\partial_x\Delta v_1\|_{L^2}\|\partial_xu_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} u_1\|_{L^2}^{\frac{1}{2}}\|\partial_x v_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xx}v_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_x\Delta v_1\|_{L^2}\|\partial_xu_2\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} u_2\|_{L^2}^{\frac{1}{2}}\|\partial_y v_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xy}v_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\nabla\Delta v\|_{L^2}\|\nabla \omega\|_{L^2}^{\frac{1}{2}}\|\Delta v\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{96}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2), \end{eqnarray*}$

于是

$J_{61}\leq \frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2).$

类似地

$J_{62}\leq \frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

$J_{63}\leq \frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

$J_{64}\leq \frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

所以

$\begin{equation} J_6\leq \frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2). \end{equation}$

(2.18)

接下来考虑 $J_7$ .

$\begin{eqnarray*} \nonumber J_7& = &\int_{{{\Bbb R}} ^2}\partial_x(v_1\partial_x u_1+v_2\partial_y u_1)\partial_x\Delta v_1{\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}\partial_x(v_1\partial_x u_2+v_2\partial_y u_2)\partial_x\Delta v_2 {\rm d}x{\rm d}y\\ \nonumber &&+\int_{{{\Bbb R}} ^2}\partial_y(v_1\partial_x u_1+v_2\partial_y u_1)\partial_y\Delta v_1{\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}\partial_y(v_1\partial_x u_2+v_2\partial_y u_2)\partial_y\Delta v_2{\rm d}x{\rm d}y\\ \nonumber& = :& J_{71}+ J_{72}+ J_{73}+J_{74}, \end{eqnarray*}$

这里

$\begin{eqnarray} J_{71}& = &\int_{{{\Bbb R}} ^2}(v_1\partial_{xx}u_1+v_2\partial_{xy}u_1) \partial_x\Delta v_1 {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}(\partial_x v_1\partial_xu_1+\partial_x v_2\partial_y u_1)\partial_x\Delta v_1{\rm d}x{\rm d}y\\ & = :& J_{711}+ J_{712}, \end{eqnarray}$

我们有

$\begin{eqnarray*} \nonumber J_{711}&\leq&C\|\partial_x\Delta v_1\|_{L^2} \|v_1\|_{L^2}^{\frac{1}{2}}\|\partial_x v_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xx} u_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xxy}u_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber&&+C\|\partial_x\Delta v_1\|_{L^2} \|v_2\|_{L^2}^{\frac{1}{2}}\|\partial_x v_2\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} u_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xyy}u_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq&C\|\nabla\Delta v\|_{L^2}\|\nabla \omega\|_{L^2}^{\frac{1}{2}} \big(\|\partial_{xx} \omega\|_{L^2}^{\frac{1}{2}}+\|\partial_{yy} \omega\|_{L^2}^{\frac{1}{2}}\big)\\ \nonumber &\leq&\frac{1}{56}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{56}\|\partial_{yy} \omega\|_{L^2}^2+\frac{1}{96}\|\nabla\Delta v\|_{L^2}^2+C\|\nabla \omega\|_{L^2}^2, \\ J_{712}&\leq&C\|\partial_x\Delta v_1\|_{L^2} \|\partial_xv_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} v_1\|_{L^2}^{\frac{1}{2}}\|\partial_x u_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xx}u_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_x\Delta v_1\|_{L^2} \|\partial_xv_2\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} v_2\|_{L^2}^{\frac{1}{2}}\|\partial_y u_1\|_{L^2}^{\frac{1}{2}}\|\partial_{xy}u_1\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\nabla\Delta v\|_{L^2}\|\Delta v\|_{L^2}^{\frac{1}{2}}\|\nabla \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{96}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2), \end{eqnarray*}$

于是

$J_{71}\leq \frac{1}{56}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{56}\|\partial_{yy} \omega\|_{L^2}^2+\frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2).$

同样地

$J_{72}\leq \frac{1}{56}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{56}\|\partial_{yy} \omega\|_{L^2}^2+\frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

$J_{73}\leq \frac{1}{56}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{56}\|\partial_{yy} \omega\|_{L^2}^2+\frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

$J_{74}\leq \frac{1}{56}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{56}\|\partial_{yy} \omega\|_{L^2}^2+\frac{1}{48}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

得到

$\begin{equation} J_7\leq \frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{14}\|\partial_{yy} \omega\|_{L^2}^2+\frac{1}{12}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2). \end{equation}$

(2.19)

对于 $J_8$ , 有

$\begin{eqnarray} J_8& = &\int_{{{\Bbb R}} ^2}\partial_x(u_1\partial_x \theta+u_2\partial_y \theta)\partial_x\Delta \theta {\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}\partial_y(u_1\partial_x \theta+u_2\partial_y \theta)\partial_y\Delta \theta {\rm d}x{\rm d}y\\ & = :& J_{81}+ J_{82}, \end{eqnarray}$

其中

$\begin{eqnarray*} \nonumber J_{81}& = &-\int_{{{\Bbb R}} ^2}\partial_{xx}(u_1\partial_x \theta+u_2\partial_y \theta)\Delta \theta {\rm d}x{\rm d}y\\ \nonumber & = &-\int_{{{\Bbb R}} ^2}u_1\partial_{xxx}\theta\Delta \theta {\rm d}x{\rm d}y-\int_{{{\Bbb R}} ^2}u_2\partial_{xxy}\theta\Delta \theta {\rm d}x{\rm d}y -2\int_{{{\Bbb R}} ^2}\partial_xu_1\partial_{xx}\theta\Delta \theta {\rm d}x{\rm d}y\\ \nonumber&&-\int_{{{\Bbb R}} ^2}(2\partial_xu_2\partial_{xy}\theta +\partial_{xx}u_1\partial_x\theta)\Delta \theta {\rm d}x{\rm d}y -\int_{{{\Bbb R}} ^2}\partial_{xx}u_2\partial_y\theta\Delta \theta {\rm d}x{\rm d}y\\ \nonumber& = :&J_{811}+ J_{812}+ J_{813}+ J_{814}+ J_{815}, \end{eqnarray*}$

又

$\begin{eqnarray*} \nonumber J_{811}& = &-\int_{{{\Bbb R}} ^2}u_1\partial_{xxx}\theta(\partial_{xx}\theta +\partial_{yy}\theta){\rm d}x{\rm d}y\\ \nonumber& = &\int_{{{\Bbb R}} ^2}\big[-\frac{1}{2}\partial_y u_2(\partial_{xx}\theta)^2 +\partial_{xx}\theta(u_1\partial_{xyy}\theta + \partial_x u_1\partial_{yy}\theta)\big]{\rm d}x{\rm d}y\\ \nonumber& = &\int_{{{\Bbb R}} ^2}(u_2\partial_{xx}\theta\partial_{xxy}\theta+ u_1\partial_{xx}\theta\partial_{xyy}\theta){\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}\partial_x u_1\partial_{xx}\theta\partial_{yy}\theta {\rm d}x{\rm d}y\\ \nonumber&\leq&C\|\partial_y\Delta \theta\|_{L^2}\|\partial_{xx}\theta\|_{L^2}^{\frac{1}{2}} \|\partial_{xxy}\theta\|_{L^2}^{\frac{1}{2}} \big(\|u_2\|_{L^2}^{\frac{1}{2}}\|\partial_x u_2\|_{L^2}^{\frac{1}{2}}+ \|u_1\|_{L^2}^{\frac{1}{2}}\|\partial_x u_1\|_{L^2}^{\frac{1}{2}}\big)\\ \nonumber &&+C\|\partial_x u_1\|_{L^2}\|\partial_{xx}\theta\|_{L^2}^{\frac{1}{2}} \|\partial_{xxy}\theta\|_{L^2}^{\frac{1}{2}} \|\partial_{yy}\theta\|_{L^2}^{\frac{1}{2}} \|\partial_{xyy}\theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq& C\|\partial_y\Delta \theta\|_{L^2}^{\frac{3}{2}}\|\Delta \theta\|_{L^2}^{\frac{1}{2}} +C\|\partial_y\Delta \theta\|_{L^2}\|\Delta \theta\|_{L^2}\\ \nonumber&\leq&\frac{1}{20}\|\partial_y\Delta \theta\|_{L^2}^2+C\|\Delta \theta\|_{L^2}^2, \\ J_{812}&\leq&C\|\partial_{xxy} \theta\|_{L^2}\|u_2\|_{L^2}^{\frac{1}{2}} \|\partial_x u_2\|_{L^2}^{\frac{1}{2}} \|\Delta \theta\|_{L^2}^{\frac{1}{2}}\|\partial_y\Delta \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq& C\|\partial_y\Delta \theta\|_{L^2}^{\frac{3}{2}}\|\Delta \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{20}\|\partial_y\Delta \theta\|_{L^2}^2+C\|\Delta \theta\|_{L^2}^2, \\ J_{813}& = & 2\int_{{{\Bbb R}} ^2}\partial_yu_2\partial_{xx}\theta\Delta \theta {\rm d}x{\rm d}y = -2\int_{{{\Bbb R}} ^2}u_2(\partial_{xx}\theta \partial_y\Delta \theta+\partial_{xxy}\theta\Delta \theta){\rm d}x{\rm d}y\\ \nonumber&\leq&C\|u_2\|_{L^2}^{\frac{1}{2}}\|\partial_x u_2\|_{L^2}^{\frac{1}{2}}\|\partial_y\Delta \theta\|_{L^2} \|\partial_{xx}\theta\|_{L^2}^{\frac{1}{2}}\|\partial_{xxy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber&&+C\|u_2 \|_{L^2}^{\frac{1}{2}}\|\partial_x u_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xxy}\theta\|_{L^2}\|\Delta \theta\|_{L^2}^{\frac{1}{2}}\|\partial_y\Delta \theta \|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\partial_y\Delta \theta\|_{L^2}^{\frac{3}{2}}\|\Delta \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{20}\|\partial_y\Delta \theta\|_{L^2}^2+C\|\Delta \theta\|_{L^2}^2, \\ J_{814}& = & -3\int_{{{\Bbb R}} ^2} \partial_xu_2\partial_{xy}\theta\Delta \theta {\rm d}x{\rm d}y -\int_{{{\Bbb R}} ^2}\partial_x u_2\partial_x\theta\partial_y\Delta \theta {\rm d}x{\rm d}y\\ \nonumber &\leq&C\|\partial_x u_2\|_{L^2}\|\partial_{xy}\theta\|_{L^2}^{\frac{1}{2}} \|\partial_{xxy}\theta\|_{L^2}^{\frac{1}{2}} \|\Delta\theta\|_{L^2}^{\frac{1}{2}} \|\partial_y\Delta\theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_y\Delta \theta\|_{L^2}\|\partial_x u_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xx}u_2\|_{L^2}^{\frac{1}{2}} \|\partial_x \theta\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq&C\|\partial_y\Delta \theta\|_{L^2}\|\Delta \theta\|_{L^2}+C\|\partial_y\Delta \theta\|_{L^2}\|\nabla \omega\|_{L^2}^{\frac{1}{2}}\|\Delta \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{20}\|\partial_y\Delta \theta\|_{L^2}^2+C(\|\Delta \theta\|_{L^2}^2+\|\nabla \omega\|_{L^2}^2), \\ J_{815}& = & \int_{{{\Bbb R}} ^2} \theta\partial_{xxy}u_2\Delta\theta {\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\theta\partial_{xx}u_2\partial_y\Delta\theta {\rm d}x{\rm d}y\\ \nonumber&\leq&C\|\partial_{xxy}u_2\|_{L^2} \|\theta\|_{L^2}^{\frac{1}{2}} \|\partial_x\theta\|_{L^2}^{\frac{1}{2}} \|\Delta\theta\|_{L^2}^{\frac{1}{2}}\|\partial_y\Delta\theta\| _{L^2}^{\frac{1}{2}}\\ \nonumber&&+C\|\partial_y\Delta \theta\|_{L^2}\|\partial_{xx} u_2\|_{L^2}^{\frac{1}{2}} \|\partial_{xxy}u_2\|_{L^2}^{\frac{1}{2}} \|\theta\|_{L^2}^{\frac{1}{2}}\|\partial_x \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber&\leq&C\|\Delta \omega\|_{L^2}\|\Delta \theta\|_{L^2}^{\frac{1}{2}}\|\partial_y\Delta\theta\| _{L^2}^{\frac{1}{2}}+C\|\partial_y\Delta \theta\|_{L^2} \|\nabla \omega\|_{L^2}^{\frac{1}{2}}\|\Delta \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{14}\|\partial_{xx}\omega\|_{L^2}^2 +\frac{1}{14}\|\partial_{yy}\omega\|_{L^2}^2 +\frac{1}{20}\|\partial_y\Delta \theta\|_{L^2}^2+C(\|\Delta \theta\|_{L^2}^2+\|\nabla \omega\|_{L^2}^2), \end{eqnarray*}$

得到

$J_{81}\leq \frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{14}\|\partial_{yy}\omega\|_{L^2}^2 +\frac{1}{4}\|\partial_y\Delta \theta\|_{L^2}^2+C(\|\Delta \theta\|_{L^2}^2+\|\nabla \omega\|_{L^2}^2).$

同时

$\begin{eqnarray} J_{82}& = &\int_{{{\Bbb R}} ^2}(u_1\partial_{xy}\theta+u_2\partial_{yy}\theta) \partial_y\Delta \theta {\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}(\partial_yu_1\partial_x\theta+\partial_yu_2 \partial_y\theta)\partial_y\Delta \theta {\rm d}x{\rm d}y\\ & = :& J_{821}+ J_{822}, \end{eqnarray}$

而

$\begin{eqnarray*} \nonumber J_{821}&\leq&C\|\partial_y\Delta \theta\|_{L^2}\|u_1\|_{L^2}^{\frac{1}{2}} \|\partial_x u_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} \theta\|_{L^2}^{\frac{1}{2}}\|\partial_{xyy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_y\Delta \theta\|_{L^2}\|u_2\|_{L^2}^{\frac{1}{2}}\|\partial_x u_2\|_{L^2}^{\frac{1}{2}} \|\partial_{yy} \theta\|_{L^2}^{\frac{1}{2}}\|\partial_{yyy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\partial_y\Delta \theta\|_{L^2}^{\frac{3}{2}}\|\Delta \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{8}\|\partial_y\Delta \theta\|_{L^2}^2+C\|\Delta \theta\|_{L^2}^2, \\ J_{822}&\leq&C\|\partial_y\Delta \theta\|_{L^2}\|\partial_yu_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} u_1\|_{L^2}^{\frac{1}{2}} \|\partial_x \theta\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_y\Delta \theta\|_{L^2}\|\partial_yu_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} u_2\|_{L^2}^{\frac{1}{2}} \|\partial_y \theta\|_{L^2}^{\frac{1}{2}}\|\partial_{yy} \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\partial_y\Delta \theta\|_{L^2}\|\nabla \omega\|_{L^2}^{\frac{1}{2}} \|\Delta \theta\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{8}\|\partial_y\Delta \theta\|_{L^2}^2+C(\|\Delta \theta\|_{L^2}^2+\|\nabla \omega\|_{L^2}^2), \end{eqnarray*}$

有

$J_{82}\leq \frac{1}{4}\|\partial_y\Delta \theta\|_{L^2}^2+C(\|\Delta \theta\|_{L^2}^2+\|\nabla \omega\|_{L^2}^2).$

所以

$\begin{equation} J_8\leq \frac{1}{14}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{14}\|\partial_{yy}\omega\|_{L^2}^2+\frac{1}{2}\|\partial_y\Delta \theta\|_{L^2}^2+C(\|\Delta \theta\|_{L^2}^2+\|\nabla \omega\|_{L^2}^2). \end{equation}$

(2.20)

综合估计式(2.13)–(2.20), 我们有

$\begin{eqnarray} &&\frac{\rm d}{{\rm d}t}\|(\nabla \omega, \Delta v, \Delta\theta )\|_{L^2}^2+\|\partial_{xx} \omega\|_{L^2}^2 +\|\partial_{yy} \omega\|_{L^2}^2 + \|\nabla\Delta v\|_{L^2}^2 +\|\partial_y \Delta \theta\|_{L^2}^2\\ &\leq&C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2+\|\Delta \theta\|_{L^2}^2). \end{eqnarray}$

(2.21)

再根据Gronwall不等式, 有

$\begin{eqnarray} &&\|(\nabla \omega, \Delta v, \Delta\theta)\|_{L^2}^2 (t) + \int_0^t (\|\partial_{xx} \omega\|_{L^2}^2 +\|\partial_{yy} \omega\|_{L^2}^2 + \|\nabla\Delta v\|_{L^2}^2 +\|\partial_y \Delta \theta\|_{L^2}^2)\ {\rm d}\tau\\ &\leq&C(t, \|(u_0, v_0, \theta_0)\|_{H^2}). \end{eqnarray}$

(2.22)

从而完成了定理1.1的证明.

3 定理1.2的证明

结合方程 $(1.3)_1$ 和 $(1.3)_2$ , 涡量 $\omega = \partial_xu_2-\partial_yu_1$ 的方程为

$\begin{equation} \partial_t \omega+u\cdot\nabla \omega-\partial_{xxx}u_2+\partial_{xxy}u_1+\partial_{xx}(v_1v_2) +\partial_{xy}v_2^2-\partial_{xy}v_1^2-\partial_{yy}(v_1v_2) = 0. \end{equation}$

(3.1)

应用分部积分法, 我们有

$\int_{{{\Bbb R}} ^2}(-\partial_{xxx}u_2+\partial_{xxy}u_1)\omega {\rm d}x{\rm d}y = \|\partial_x \omega\|_{L^2}^2.$

与定理1.1的证明类似, 得到解 $(u, v, \theta)$ 的 $L^2$ 估计和 $H^1$ 估计如下:

$\begin{eqnarray} &&\|(u, v, \theta)\|_{L^2}^2 (t) + 2\int_0^t (\|\partial_x u_1\|_{L^2}^2 + \|\partial_x u_2\|_{L^2}^2 + \|\nabla v\|_{L^2}^2+ \|\partial_y \theta\|_{L^2}^2)\ {\rm d}\tau\\ & = & \|(u_0, v_0, \theta_0)\|_{L^2}^2, \end{eqnarray}$

(3.2)

$\begin{eqnarray} &&\|(\omega, \nabla v, \nabla\theta)\|_{L^2}^2 (t) + \int_0^t (\|\partial_x \omega\|_{L^2}^2+ \|\Delta v\|_{L^2}^2+ \|\partial_y \nabla\theta\|_{L^2}^2)\ {\rm d}\tau \\ &\leq& C(t, \|(u_0, v_0, \theta_0)\|_{H^1}), \end{eqnarray}$

(3.3)

对任意 $t>0$ 成立.

接下来我们证明 $\|(u, v, \theta)\|_{H^2}$ 有界.

用算子 $\nabla$ 作用于方程 $(3.1), (1.3)_4, (1.3)_5$ 的两边, 然后分别用 $\nabla \omega, -\nabla\Delta v, -\nabla\Delta \theta$ 作 $L^2$ 内积, 有

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\|(\nabla \omega, \Delta v, \Delta\theta )\|_{L^2}^2+\|\partial_{xx} \omega\|_{L^2}^2 +\|\partial_{xy} \omega\|_{L^2}^2 + \|\nabla\Delta v\|_{L^2}^2 +\|\partial_y \Delta \theta\|_{L^2}^2\\ & = &\int_{{{\Bbb R}} ^2}\partial_{xx}(v_1v_2)\Delta \omega {\rm d}x{\rm d}y+ \int_{{{\Bbb R}} ^2}\partial_{xy}v_2^2 \Delta \omega {\rm d}x{\rm d}y -\int_{{{\Bbb R}} ^2}\partial_{xy}v_1^2 \Delta \omega {\rm d}x{\rm d}y\\ &&-\int_{{{\Bbb R}} ^2}\partial_{yy}(v_1v_2) \Delta \omega {\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}u\cdot\nabla \omega\Delta \omega {\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\nabla (u\cdot\nabla) v:\nabla\Delta v{\rm d}x{\rm d}y\\ &&+\int_{{{\Bbb R}} ^2}\nabla(v\cdot\nabla) u:\nabla\Delta v{\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}\nabla (u\cdot\nabla) v\cdot\nabla\Delta \theta {\rm d}x{\rm d}y\\ & = :& K_1+K_2+K_3+K_4+K_5+K_6+K_7+K_8, \end{eqnarray}$

这里我们利用下面的等式

$\int_{{{\Bbb R}} ^2}\nabla(-\partial_{xxx}u_2+\partial_{xxy}u_1)\cdot\nabla \omega {\rm d}x{\rm d}y = \|\partial_{xx}\omega\|_{L^2}^2 +\|\partial_{xy}\omega\|_{L^2}^2,$

$\int_{{{\Bbb R}} ^2}\nabla\nabla\theta:\nabla\Delta v{\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\nabla(\nabla\cdot v) \cdot\nabla\Delta\theta {\rm d}x{\rm d}y = 0.$

其中 $K_6, K_7$ 和 $K_8$ 的估计, 与定理1.1的证明类似, 其结果如下:

$\begin{equation} K_6\leq \frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2), \end{equation}$

(3.4)

$\begin{equation} K_7\leq \frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{12}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{12}\|\nabla\Delta v\|_{L^2}^2+C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2), \end{equation}$

(3.5)

$\begin{equation} K_8\leq \frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2 +\frac{1}{12}\|\partial_{xy}\omega\|_{L^2}^2+\frac{1}{2}\|\partial_y \Delta \theta\|_{L^2}^2+C(\|\Delta \theta\|_{L^2}^2+\|\nabla \omega\|_{L^2}^2). \end{equation}$

(3.6)

下面我们用部分积分法、各向异性的Sobolev不等式和Young不等式分别估计 $K_1, K_2,$ $K_3, K_4$ 和 $K_5$ .

$\begin{eqnarray} K_1& = &\int_{{{\Bbb R}} ^2}v_1\partial_{xx}v_2 \Delta \omega {\rm d}x{\rm d}y + 2\int_{{{\Bbb R}} ^2}\partial_x v_1\partial_x v_2 \Delta \omega {\rm d}x{\rm d}y + \int_{{{\Bbb R}} ^2}\partial_{xx}v_1v_2 \Delta \omega {\rm d}x{\rm d}y\\ & = :& K_{11}+ K_{12}+ K_{13}, \end{eqnarray}$

其中

$\begin{eqnarray} K_{11}& = &\int_{{{\Bbb R}} ^2}v_1\partial_{xx}v_2\partial_{xx} \omega {\rm d}x{\rm d}y-\int_{{{\Bbb R}} ^2}\partial_y(v_1\partial_{xx}v_2) \partial_y \omega {\rm d}x{\rm d}y\\ & = &\int_{{{\Bbb R}} ^2}v_1\partial_{xx}v_2\partial_{xx} \omega {\rm d}x{\rm d}y-\int_{{{\Bbb R}} ^2}v_1\partial_{xxy}v_2 \partial_y \omega {\rm d}x{\rm d}y\\ &&+\int_{{{\Bbb R}} ^2}\partial_yv_1\partial_{xxy}v_2 \omega {\rm d}x{\rm d}y+\int_{{{\Bbb R}} ^2}\partial_{yy}v_1\partial_{xx}v_2 \omega {\rm d}x{\rm d}y\\ & = :& K_{111}+ K_{112}+ K_{113}+ K_{114}, \end{eqnarray}$

同时

$\begin{eqnarray*} \nonumber K_{111}&\leq&C\|\partial_{xx} \omega\|_{L^2}\|v_1\|_{L^2}^{\frac{1}{2}} \|\partial_xv_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xx} v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xxy} v_2\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\partial_{xx} \omega\|_{L^2}\|\Delta v\|_{L^2}^{\frac{1}{2}}\|\nabla\Delta v\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{36}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{144}\|\nabla\Delta v\|_{L^2}^2 +C\|\Delta v\|_{L^2}^2, \\ K_{112}&\leq&C\|\partial_{xxy} v_2\|_{L^2}\|v_1\|_{L^2}^{\frac{1}{2}} \|\partial_yv_1\|_{L^2}^{\frac{1}{2}} \|\partial_y \omega\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\nabla\Delta v\|_{L^2}\|\Delta \omega\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}\omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{24}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{144}\|\nabla\Delta v\|_{L^2}^2 +C\|\nabla \omega\|_{L^2}^2, \\ K_{113}&\leq&C\|\partial_{xxy} v_2\|_{L^2}\|\partial_yv_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}v_1\|_{L^2}^{\frac{1}{2}} \|\omega\|_{L^2}^{\frac{1}{2}}\|\partial_y \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\nabla\Delta v\|_{L^2}\|\Delta v\|_{L^2}^{\frac{1}{2}}\|\nabla \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{144}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2), \\ K_{114}&\leq&C\|\omega\|_{L^2}\|\partial_{yy}v_1\| _{L^2}^{\frac{1}{2}}\|\partial_{xyy}v_1\|_{L^2}^{\frac{1}{2}} \|\partial_{xx} v_2\|_{L^2}^{\frac{1}{2}}\|\partial_{xxy} v_2\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\nabla\Delta v\|_{L^2}\|\Delta v\|_{L^2}\\ \nonumber &\leq&\frac{1}{144}\|\nabla\Delta v\|_{L^2}^2 +C\|\Delta v\|_{L^2}^2, \end{eqnarray*}$

有

$K_{11}\leq \frac{1}{36}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{24}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{36}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2).$

类似地

$K_{12}\leq \frac{1}{36}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{36}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

$K_{13}\leq \frac{1}{36}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{24}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{36}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2),$

因此

$\begin{equation} K_1\leq \frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{12}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{12}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2). \end{equation}$

(3.7)

同样地, 对于 $K_2, K_3, K_4$ , 我们有

$\begin{equation} K_2\leq \frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{12}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{12}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2), \end{equation}$

(3.8)

$\begin{equation} K_3\leq \frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{12}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{12}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2), \end{equation}$

(3.9)

$\begin{equation} K_4\leq \frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{12}\|\partial_{xy} \omega\|_{L^2}^2+\frac{1}{12}\|\nabla\Delta v\|_{L^2}^2 +C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2). \end{equation}$

(3.10)

最后我们估计 $K_5$ .

$\begin{eqnarray} K_5& = &\int_{{{\Bbb R}} ^2}(u_1\partial_x \omega+u_2\partial_y \omega)\partial_{xx}\omega {\rm d}x{\rm d}y +\int_{{{\Bbb R}} ^2}(u_1\partial_x \omega+u_2\partial_y \omega)\partial_{yy}\omega {\rm d}x{\rm d}y\\ & = :&K_{51}+K_{52}, \end{eqnarray}$

其中

$\begin{eqnarray*} \nonumber K_{51}&\leq&C\|\partial_{xx} \omega\|_{L^2} \|u_1\|_{L^2}^{\frac{1}{2}}\|\partial_y u_1\|_{L^2}^{\frac{1}{2}} \|\partial_x \omega\|_{L^2}^{\frac{1}{2}} \|\partial_{xx}\omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_{xx} \omega\|_{L^2} \|u_2\|_{L^2}^{\frac{1}{2}}\|\partial_y u_2\|_{L^2}^{\frac{1}{2}} \|\partial_y \omega\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}\omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\partial_{xx} \omega\|_{L^2}^{\frac{3}{2}}\|\nabla \omega\|_{L^2}^{\frac{1}{2}}+C\|\partial_{xx} \omega\|_{L^2} \|\nabla \omega\|_{L^2}^{\frac{1}{2}}\|\partial_{xy} \omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq&\frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{24}\|\partial_{xy} \omega\|_{L^2}^2+C\|\nabla \omega\|_{L^2}^2, \\ K_{52}& = &\int_{{{\Bbb R}} ^2}\big[-\partial_y(u_1\partial_x \omega)\partial_y \omega+\frac{1}{2}\partial_xu_1(\partial_y \omega)^2\big]{\rm d}x{\rm d}y\\ \nonumber & = & -2\int_{{{\Bbb R}} ^2}u_1\partial_{xy} \omega\partial_y \omega {\rm d}x{\rm d}y-\int_{{{\Bbb R}} ^2}\partial_yu_1\partial_x \omega\partial_y \omega {\rm d}x{\rm d}y\\ \nonumber&\leq&C\|\partial_{xy}\omega\|_{L^2} \|u_1\|_{L^2}^{\frac{1}{2}}\|\partial_y u_1\|_{L^2}^{\frac{1}{2}} \|\partial_y \omega\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}\omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &&+C\|\partial_y u_1\|_{L^2}\|\partial_x \omega\|_{L^2}^{\frac{1}{2}} \|\partial_{xy} \omega\|_{L^2}^{\frac{1}{2}} \|\partial_y \omega\|_{L^2}^{\frac{1}{2}} \|\partial_{xy}\omega\|_{L^2}^{\frac{1}{2}}\\ \nonumber &\leq& C\|\partial_{xy} \omega\|_{L^2}^{\frac{3}{2}}\|\nabla \omega\|_{L^2}^{\frac{1}{2}}+C\|\partial_{xy} \omega\|_{L^2} \|\nabla \omega\|_{L^2}\\ \nonumber &\leq&\frac{1}{24}\|\partial_{xy} \omega\|_{L^2}^2+C\|\nabla \omega\|_{L^2}^2, \end{eqnarray*}$

从而

$\begin{equation} K_5\leq \frac{1}{12}\|\partial_{xx} \omega\|_{L^2}^2+\frac{1}{12}\|\partial_{xy} \omega\|_{L^2}^2+C\|\nabla \omega\|_{L^2}^2. \end{equation}$

(3.11)

综合估计式(3.4)–(3.11), 可得

$\begin{eqnarray} &&\frac{\rm d}{{\rm d}t}\|(\nabla \omega, \Delta v, \Delta\theta )\|_{L^2}^2+\|\partial_{xx} \omega\|_{L^2}^2 +\|\partial_{xy} \omega\|_{L^2}^2 + \|\nabla\Delta v\|_{L^2}^2 +\|\partial_y \Delta \theta\|_{L^2}^2\\ &\leq&C(\|\nabla \omega\|_{L^2}^2+\|\Delta v\|_{L^2}^2+\|\Delta \theta\|_{L^2}^2). \end{eqnarray}$

(3.12)

根据Gronwall不等式, 有

$\begin{eqnarray} &&\|(\nabla \omega, \Delta v, \Delta\theta)\|_{L^2}^2 (t) + \int_0^t (\|\partial_{xx} \omega\|_{L^2}^2 +\|\partial_{xy} \omega\|_{L^2}^2 + \|\nabla\Delta v\|_{L^2}^2 +\|\partial_y \Delta \theta\|_{L^2}^2)\ {\rm d}\tau\\ &\leq&C(t, \|(u_0, v_0, \theta_0)\|_{H^2}), \end{eqnarray}$

(3.13)

对任意 $t>0$ 成立. 从而完成了定理1.2的证明.

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

[1]

Frierson

, Majda

, Pauluis

Large scale dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit

Commun Math Sci, 2004, 2, 591- 626

DOI:10.4310/CMS.2004.v2.n4.a3 [本文引用: 1]

[2]

J K

, Titi

E S

Global well-posedness of strong solutions to a tropical climate model

Discrete Contin Dyn Syst, 2016, 36, 4495- 4516

DOI:10.3934/dcds.2016.36.4495 [本文引用: 1]

[3]

C C

, Wan

R H

Spectral analysis and global well-posedness for a viscous tropical climate model with only a damp term

Nonlinear Anal RWA, 2018, 39, 554- 567

DOI:10.1016/j.nonrwa.2017.08.004 [本文引用: 1]

[4]

Wan

R H

Global small solutions to a tropical climate model without thermal diffusion

J Math Phys, 2016, 57 (2): 1- 13

URL [本文引用: 1]

[5]

, Zhu

M X

Global strong solutions of the tropical climate model with temperature-dependent diffusion on the barotropic mode

Appl Math Lett, 2019, 89, 8- 14

DOI:10.1016/j.aml.2018.09.009 [本文引用: 1]

[6]

, Zhu

M X

Global strong solutions of the 2D tropical climate system with temperature-dependent viscosity

Z Angew Math Phys, 2020, 71, Article number: 97

DOI:10.1007/s00033-020-01321-9 [本文引用: 1]

[7]

Global regularity for a class of 2D tropical climate model

J Math Anal Appl, 2017, 446, 307- 321

DOI:10.1016/j.jmaa.2016.08.053 [本文引用: 1]

[8]

Zhu

M X

Global regularity for the tropical climate model with fractional diffusion on barotropic mode

Appl Math Lett, 2018, 81, 99- 104

DOI:10.1016/j.aml.2018.02.003 [本文引用: 1]

[9]

Dong

B Q

, Wang

W J

, Wu

J H

, et al.

Global regularity for a class of 2D generalized tropical climate models

J Differential Equations, 2019, 266, 6346- 6382

DOI:10.1016/j.jde.2018.11.007 [本文引用: 1]

[10]

Dong

B Q

, Wang

W J

, Wu

J H

, Zhang

Global regularity results for the climate model with fractional dissipation

Discrete Contin Dyn Syst, 2019, 24, 211- 229

URL [本文引用: 1]

[11]

Regmi

, Wu

J H

Global regularity for the 2D magneto-micropolar equations with partial dissipation

J Math Study, 2016, 49, 169- 194

DOI:10.4208/jms.v49n2.16.05 [本文引用: 2]

[12]

Cao

C S

, Wu

J H

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Adv in Math, 2011, 226, 1803- 1822

DOI:10.1016/j.aim.2010.08.017 [本文引用: 1]

[13]

Dong

B Q

, Wu

J H

, Ye

2D tropical climate model with fractional dissipation and without thermal diffusion

Commun Math Sci, 2020, 18, 259- 292

DOI:10.4310/CMS.2020.v18.n1.a11

[14]

Dong

B Q

, Wu

J H

, Ye

Global regularity for a 2D tropical climate model with fractional dissipation

J Nonlinear Sci, 2019, 29, 511- 550

DOI:10.1007/s00332-018-9495-5

[15]

C C

, Jiang

Z H

, Wan

R H

Local well-posedness for the tropical climate model with fractional velocity diffusion

Kinet Relat Models, 2016, 9, 551- 570

DOI:10.3934/krm.2016006

[16]

Global regularity of 2D tropical climate model with zero thermal diffusion

Z Angew Math Mech, 2020, 100, Article number: 7

URL [本文引用: 1]

Large scale dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit

2004

... Tropical Climate方程是由Frierson-Majda-Pauluis^[1]通过对完全无粘的Primitive方程中的第一斜压模型采用Galerkin截断推导出来的. 下面我们简单总结一下Tropical Climate方程现有的研究成果. Li和Titi^[2]通过引入伪斜压速度

$\omega = v-\nabla(-\Delta)^{-1}\theta$

, 并利用对数型Gronwall不等式, 证明了当正压模型和第一斜压模型具有标准的Laplace耗散项而温度场没有耗散项时2维Tropical Climate方程整体适定. Wan和Ma^[3-4]证明了2维Tropical Climate方程小初值强解整体适定, Ye和Zhu^[5-6]证明了2维Tropical Climate方程中正压模型具有与温度场相关的粘性时存在唯一的整体光滑解. 近年来, 数学家开始着手研究分数阶Tropical Climate方程, 其中耗散项分别为

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global well-posedness of strong solutions to a tropical climate model

2016

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Spectral analysis and global well-posedness for a viscous tropical climate model with only a damp term

2018

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global small solutions to a tropical climate model without thermal diffusion

2016

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global strong solutions of the tropical climate model with temperature-dependent diffusion on the barotropic mode

2019

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global strong solutions of the 2D tropical climate system with temperature-dependent viscosity

2020

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global regularity for a class of 2D tropical climate model

2017

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global regularity for the tropical climate model with fractional diffusion on barotropic mode

2018

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global regularity for a class of 2D generalized tropical climate models

2019

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global regularity results for the climate model with fractional dissipation

2019

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

Global regularity for the 2D magneto-micropolar equations with partial dissipation

2016

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

... 受文章[11]启发, 本文致力于研究2维带部分粘性Tropical Climate方程的整体存在性和正则性, 有下面几种情形. ...

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

2011

... 引理1.1^[12] 假设

$f, \ g, \ h, \ \partial_x g, \ \partial_y h\in L^2({{\Bbb R}} ^2)$

, 则 ...

2D tropical climate model with fractional dissipation and without thermal diffusion

2020

Global regularity for a 2D tropical climate model with fractional dissipation

2019

Local well-posedness for the tropical climate model with fractional velocity diffusion

2016

Global regularity of 2D tropical climate model with zero thermal diffusion

2020

$\omega = v-\nabla(-\Delta)^{-1}\theta$

$\Lambda^{2\alpha}u, \ \Lambda^{2\beta}v$

和

$\Lambda^{2\gamma}\theta$

. Ye^[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当

$\alpha>0, \ \beta = \gamma = 1$

或

$\mu = 0, \ \beta>1, \ \gamma = 1$

时2维Tropical Climate方程整体适定. Zhu^[8] 推出了当

$\alpha\geq\frac{5}{2}, \ \nu = \eta = 0$

时3维Tropical Climate方程存在唯一的整体强解. Dong等通过引入新的量

$H = \nabla\cdot v-\Lambda^{2-2\beta}\theta$

并运用对数型Sobolev不等式证明了当

$\alpha = 0, \ \beta>1, \ \beta+\gamma>\frac{3}{2}$

或

$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0$

时2维Tropical Climate方程存在唯一的整体光滑解^[9], 根据分数阶热算子的极大

$L^q_tL^p_x$

正则性，并结合Besov空间理论证明了当

$\alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$

或

$\alpha = 2, \ \beta = \gamma = 0$

时, 2维Tropical Climate方程的整体适定性^[10]. 更多结果参见文献[11-16]. ...

〈

〉