本文主要研究2维带部分粘性的Tropical Climate方程

(1.1) $ \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} $

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

(1.2) $ \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} $

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

(1.3) $ \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} $

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

(1.4) $ \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} $

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.

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

(2.1) $ \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} $

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

(2.2) $ \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} $

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

(2.3) $ \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} $

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

(2.4) $ \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} $

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

(2.5) $ \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} $

类似地

(2.6) $ \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} $

(2.7) $ \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} $

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

(2.8) $ \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} $

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

(2.9) $ \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} $

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

(2.10) $ \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} $

综合估计式(2.4)–(2.10), 可得

(2.11) $ \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} $

应用Gronwall不等式和(2.2)式, 可得

(2.12) $ \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} $

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

(2.13) $ \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} $

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

(2.14) $ \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} $

同样地

(2.15) $ \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} $

(2.16) $ \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} $

下面估计$ J_5 $.

(2.17) $ \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} $

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

(2.18) $ \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} $

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

(2.19) $ \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} $

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

(2.20) $ \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} $

综合估计式(2.13)–(2.20), 我们有

(2.21) $ \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} $

再根据Gronwall不等式, 有

(2.22) $ \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} $

从而完成了定理1.1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

(3.1) $ \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} $

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:

(3.2) $ \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} $

(3.3) $ \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} $

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:

(3.4) $ \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} $

(3.5) $ \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} $

(3.6) $ \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} $

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

(3.7) $ \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} $

同样地, 对于$ K_2, K_3, K_4 $, 我们有

(3.8) $ \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} $

(3.9) $ \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} $

(3.10) $ \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} $

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

(3.11) $ \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} $

综合估计式(3.4)–(3.11), 可得

(3.12) $ \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} $

根据Gronwall不等式, 有

(3.13) $ \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} $

对任意$ t>0 $成立. 从而完成了定理1.2的证明.