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

Wang Wenjuan,, Xue Mingxiang,

School of Mathematical Sciences, Anhui University, Hefei 230601

 基金资助: 国家自然科学基金.  12001004

 Fund supported: the NSFC.  12001004

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

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

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

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 $$\Lambda^{2\gamma}\theta . Ye[7]利用弱的非线性能量估计逼近和热算子的极大正则性证明了当 \alpha>0, \ \beta = \gamma = 1$$ \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} $$\frac{3}{2}<\beta\leq2, \ \alpha = \gamma = 0 时2维Tropical Climate方程存在唯一的整体光滑解[9], 根据分数阶热算子的极大 L^q_tL^p_x 正则性，并结合Besov空间理论证明了当 \alpha+\beta = 2, 1<\beta\leq\frac{3}{2}, \gamma = 0$$ \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].

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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