湿大气三维粘性原始方程组时间周期解的存在性

doi:10.12677/aam.2025.1411463

期刊菜单

湿大气三维粘性原始方程组时间周期解的存在性
Existence of Time Periodic Solutions for the 3D Viscous Primitive Equations of Large Scale Monist Atmosphere

DOI: 10.12677/aam.2025.1411463, PDF, HTML, XML,
作者: 罗维：武警警官学院基础部，四川成都
关键词: 湿大气三维粘性原始方程组；时间周期解；Galerkin方法；3D Viscous Primitive Equations of Monist Atmosphere； Time Periodic Solutions； Galerkin Method

摘要: 本文考虑的是在气压坐标下的湿大气三维粘性原始方程组的时间周期解的存在性和唯一性问题。主要运用了Galerkin方法，即首先利用Leray-Schauder不动点定理来证明大气原始方程组具有周期性的近似解的存在性，然后再证明这个近似解在其工作空间的收敛性，从而得到大气原始方程组的周期解的存在性，并证明了其唯一性。

Abstract: In this paper, we consider the existence of time periodic solutions of the 3D viscous primitive equations of large-scale monist atmosphere. We used the Galerkin method. Firstly, by Leray-Schauder fixed point theorem, we prove the existence of approximate solutions of the primitive equations, then we show the convergence of the approximate solutions, and we also get the uniqueness to the primitive equations.

文章引用：罗维. 湿大气三维粘性原始方程组时间周期解的存在性[J]. 应用数学进展, 2025, 14(11): 76-89. https://doi.org/10.12677/aam.2025.1411463

1. 引言

众所周知，天气预报对许多社会经济部门的发展都有非常大的贡献，与日常生活也息息相关。因此对大气相关问题的科学研究具有重大的理论意义和应用价值。Richardson在1992年，首次提出原始大气方程组(由状态方程、携带科氏力的流体力学方程、热动力学方程组成)，并提出了关于数值天气预报的概念，可参见文献[1]。20世纪60年代中期，随着计算机能力的明显增强和大气科学的高速发展，人们为了延长数值天气预报的时效转向应用原始方程组进行数值天气预报，大气原始方程组开始在数值天气预报中发挥重要作用。

2006年郭柏灵与黄代文受到Lions、Temam和Wang得到干大气原始方程组新形式用到的方法的启发，得到了气压坐标下湿大气原始方程组的新形式[2]，接着在2007年与2008年他们给出了干湿大气原始方程组新形式的简化模式[3] [4]。本文中所研究的湿大气原始方程组就是郭柏灵与黄代文所给出的湿大气原始方程组新形式的简化模式。

周期现象是自然中一种非常特殊的现象，随着长时间的推移这些现象有规律的重复，在数学中表现为非线性微分方程的周期解。本文主要研究大尺度湿大气三维粘性原方程时间周期解的存在性与唯一性。研究大气原始方程组的周期解，是希望以它的周期解作为突破口来阐明原始大气方程解的一些性质，进一步加深人们对自然界广泛存在的各种自然现象的认识和理解，为人们利用自然和改造自然提供强有力的理论基础。本文主要结果是定理3.4.1、定理3.4.2。为了证明原始方程的时间周期解，本文使用了著名的Galerkin方法，该方法用于证明许多系统的时间周期解和弱解的存在性，比如Navier-Stokes方程、Schr-odinger-Boussinesq方程、量子方程和伪抛物方程。因此，在文献[5]-[11]中证明方法的引导下，可以得到期望的结果。

2. 预备知识

2.1. 湿大气三维粘性原始方程组

气压坐标下无量纲的湿大气三维粘性原始方程组为

$\frac{\partial \vec{v}}{\partial t} + \nabla_{\vec{v}} \vec{v} + ω \frac{\partial \vec{v}}{\partial ξ} + \frac{f}{R_{0}} k \times \vec{v} + g r a d Φ - \frac{1}{{Re}_{1}} Δ \vec{v} - \frac{1}{{Re}_{2}} \frac{\partial^{2} \vec{v}}{\partial ξ^{2}} = 0$ ， (1)

$d i v \vec{v} + \frac{\partial ω}{\partial ξ} = 0$ ， (2)

$\frac{\partial Φ}{\partial ξ} + \frac{b P}{p} (1 + a q) T = 0$ ， (3)

$\frac{\partial T}{\partial t} + \nabla_{\vec{v}} T + ω \frac{\partial T}{\partial ξ} - \frac{b P}{p} (1 + a q) ω - \frac{1}{R t_{1}} Δ T - \frac{1}{R t_{2}} \frac{\partial^{2} T}{\partial ξ^{2}} = Q_{1}$ ， (4)

$\frac{\partial q}{\partial t} + \nabla_{\vec{v}} q + ω \frac{\partial q}{\partial ξ} - \frac{1}{R q_{1}} Δ q - \frac{1}{R q_{2}} \frac{\partial^{2} q}{\partial ξ^{2}} = Q_{2}$ ， (5)

其中 $\vec{v} = (v_{θ}, v_{ϕ})$ 是气压坐标下水平方向的速度， $ω$ 是气压坐标下垂直方向上的速度， $T$ 代表温度， $q$ 代表空气中水蒸气的混合比， $Φ$ 代表地势，这里 $\vec{v}, ω, Φ, T, q$ 都是未知的函数， $f = 2 \cos θ$ 是科氏参数， $R_{0}$ 是Rossby数， $k$ 是单位向量(垂直方向上)， ${Re}_{1}$ 是水平方向上的雷诺数， ${Re}_{2}$ 是垂直方向上的雷诺数， $R t_{1}$ 代表水平方向热扩散系数， $R t_{2}$ 代表垂直方向上的热扩散系数， $R q_{1}$ 代表水平方向水蒸气扩散系数， $R q_{2}$ 代表垂直方向上的水蒸气扩散系数， $p_{0}$ 是地球表面压力的近似值，P是上层大气的压力且 $p_{0} > 0$ ，变量 $ξ$ 满足 $p = (P - p_{0}) ξ + p_{0}$ $(0 < p_{0} \leq p \leq P)$ ， $Q_{1}$ 与 $Q_{2}$ 是 $S^{2} \times (0, 1)$ 上的已知函数，a是正的常数( $a = 0.618$ )，b是正的常数。

方程组(1)~(5)的空间区域为 $Ω = S^{2} \times (0, 1)$ ，其中 $S^{2}$ 是二维的单位球面。边界条件为

$ξ = 1 (p = P) : \frac{\partial \vec{v}}{\partial ξ} = 0, ω = 0, \frac{\partial T}{\partial ξ} = α_{s} (T_{s} - T), \frac{\partial q}{\partial ξ} = β_{s} (q_{s} - q)$ ， (6)

$ξ = 0 (p = P_{0}) : \frac{\partial \vec{v}}{\partial ξ} = 0, ω = 0, \frac{\partial T}{\partial ξ} = 0, \frac{\partial q}{\partial ξ} = 0$ ， (7)

其中 $α_{s}$ 与 $β_{s}$ 都是正的常数， $T_{s}$ 是地球表面的温度， $q_{s}$ 是地球表面上给定的水蒸气的混合比，本文中为了简化(不失一般性)，假设 $T_{s} = 0, q_{s} = 0$ 。

参考文献[12]，可以将方程组(1)~(5)化为

$\frac{\partial \vec{v}}{\partial t} + \nabla_{\vec{v}} \vec{v} + (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) \frac{\partial \vec{v}}{\partial ξ} + \frac{f}{R_{0}} k \times \vec{v} + g r a d Φ_{s} + \int_{ξ}^{1} \frac{b P}{p} g r a d [(1 + a q) T] d ξ^{'} - \frac{1}{{Re}_{1}} Δ \vec{v} - \frac{1}{{Re}_{2}} \frac{\partial^{2} \vec{v}}{\partial ξ^{2}} = 0,$ (8)

$\frac{\partial T}{\partial t} + \nabla_{\vec{v}} T + (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) \frac{\partial T}{\partial ξ} - \frac{b P}{p} (1 + a q) (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) - \frac{1}{R t_{1}} Δ T - \frac{1}{R t_{2}} \frac{\partial^{2} T}{\partial ξ^{2}} = Q_{1},$ (9)

$\frac{\partial q}{\partial t} + \nabla_{\vec{v}} q + (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) \frac{\partial q}{\partial ξ} - \frac{1}{R q_{1}} Δ q - \frac{1}{R q_{2}} \frac{\partial^{2} q}{\partial ξ^{2}} = Q_{2} ，$ (10)

$\int_{0}^{1} d i v \vec{v} d ξ = 0.$ (11)

方程组(8)~(11)的边界条件为

$ξ = 1 : \frac{\partial \vec{v}}{\partial ξ} = 0, \frac{\partial T}{\partial ξ} = - α_{s} T, \frac{\partial q}{\partial ξ} = - β_{s} q,$ (12)

$ξ = 0 : \frac{\partial \vec{v}}{\partial ξ} = 0, \frac{\partial T}{\partial ξ} = 0, \frac{\partial q}{\partial ξ} = 0.$ (13)

本文考虑的问题如下：使给定的函数 $Q_{1}, Q_{2}$ 随着时间的变化是W为周期的周期函数，证明在 $K \equiv \sup_{0 \leq t \leq W} {‖ Q_{1} ‖}_{L^{N} (Ω)}$ ， $K^{'} \equiv \sup_{0 \leq t \leq W} {‖ Q_{2} ‖}_{L^{N} (Ω)}$ 足够小的情况下，湿大气三维粘性原始方程组时间周期解存在，且周期为W。由于将方程组(1)~(5)化简为了(8)~(11)，所以只需证明方程组(8)~(11)的解以W为周期，即

$\begin{matrix} U (t + W) = (\vec{v} (t + W), T (t + W), q (t + W)) \\ = (\vec{v}, T, q) = U (t) x \in Ω, t \in R^{1} . \end{matrix}$ (14)

2.2. 一些函数空间

首先介绍一些将会用到的函数空间和算子。对于二位球面，坐标为 $(θ, ϕ)$ 的任意点的位置矢量是 $\vec{R} = \sin θ \cos ϕ \vec{i} + \sin θ \sin ϕ \vec{j} + \cos θ \vec{k}$ 。 $e_{θ}, e_{ϕ}, e_{ξ}$ 分别为空间域 $Ω$ 中 $θ, ϕ, ξ$ 方向上的单位矢量，且

$e_{θ} = \frac{\partial}{\partial θ}, e_{ϕ} = \frac{1}{\sin θ} \frac{\partial}{\partial ϕ}, e_{ξ} = \frac{\partial}{\partial ξ} .$

空间 $L^{p} (Ω) : = {h; h : Ω \to R, \int_{Ω} {| h |}^{p} d Ω < + \infty}$ 范数是 ${| h |}_{p} = {(\int_{Ω} {| h |}^{p} d Ω)}^{\frac{1}{p}}$ ， $1 \leq p < \infty$ (本文中有时将 $\int_{Ω} \cdot d Ω, \int_{S^{2}} \cdot d S^{2}$ 简写为 $\int_{Ω} \cdot, \int_{S^{2}} \cdot$ )。 $L^{2} (T Ω | T S^{2})$ 是由 $Ω$ 中的 $L^{2}$ 向量场的前两个分量组成的空间，范数为 ${| \vec{v} |}_{2} = {(\int_{Ω} (v_{θ}^{2} + v_{ϕ}^{2}) d Ω)}^{1 / 2}$ ，这里 $\vec{v} = (v_{θ}, v_{ϕ}) : Ω \to T S^{2}$ 。 $C^{\infty} (S^{2})$ 是由 $S^{2}$ 到 $R$ 的光滑函数全体所组成的空间。 $C^{\infty} (Ω)$ 是由 $Ω$ 到 $R$ 光滑函数全体所组成的空间。 $C^{\infty} (T Ω | T S^{2})$ 是由 $Ω$ 中的光滑向量场的前面两个分量组成的空间。 $C_{0}^{\infty} (Ω) : = {h; h \in C^{\infty} (Ω), supp h 是 Ω 的 � 子集}$ 。 $C_{0}^{\infty} (T Ω | T S^{2}) = {\vec{v}; \vec{v} \in C^{\infty} (T Ω | T S^{2}),$ $s u p p {\vec{v}}^{} 是 Ω 的 � 子集}$ 。 $H^{m} (Ω)$ 是由本身及其所有关于 $e_{θ}, e_{ϕ}, e_{ξ}$ 小于等于m阶的协变导数都在 $L^{2}$ 中的函数所组成的Sobolev空间，其中的范数为 ${‖ h ‖}_{m} = {(\int_{Ω} ({\sum_{1 \leq k \leq m} \sum_{i_{j} = 1, 2, 3; j = 1, \dots, k} | \nabla_{i_{1}} \cdot \cdot \cdot \nabla_{i_{k}} h |}^{2} + {| h |}^{2}))}^{\frac{1}{2}}$ ，这里的 $\nabla_{1} = \nabla_{e_{θ}}$ ， $\nabla_{2} = \nabla_{e_{ϕ}}$ ， $\nabla_{3} = \nabla_{e_{ξ}} = \frac{\partial}{\partial ξ}$ (将在后面给出协变导数算子 $\nabla_{e_{θ}}, \nabla_{e_{ϕ}}$ 的定义)。 $H^{\infty} (T Ω | T S^{2}) =$ ${\vec{v}; \vec{v} = (v_{θ}, v_{ϕ}) : Ω \to T S^{2}, v_{θ}, v_{ϕ} \in H^{m} (Ω)}$ ， $H^{\infty} (T Ω | T S^{2})$ 中的范数与 $H^{m} (Ω)$ 中的范数非常的相似，即在 $H^{m} (Ω)$ 的范数公式中使 $h = (v_{θ}, v_{ϕ}) = v_{θ} e_{θ} + v_{ϕ} e_{ϕ}$ 。

然后介绍一些以W为周期的函数构成的函数空间。令X是巴拿赫空间，定义空间 $C^{k} (W; X)$ 是在 $R^{1}$ 中以W为周期的函数的k阶导在X中连续的集合，定义范数

${| Q |}_{C^{k} (W; X)} = \sup_{0 \leq t \leq W} {\sum_{i = 0}^{k} {| D_{t}^{i} Q (t) |}_{X}} .$

$L^{p} (W; X)$ $(1 \leq p \leq \infty)$ 是在 $R^{1}$ 中以W为周期且在X上可测的函数的集合，定义范数

${| Q |}_{L^{p} (W; X)} = {(\int_{0}^{W} {| Q |}_{X}^{p} d t)}^{\frac{1}{p}} < + \infty (1 \leq p < \infty)$ ，

${| Q |}_{L^{\infty} (W; X)} = \sup_{0 \leq t \leq W} {| Q |}_{X} < + \infty .$

$W^{k, p} (W; X) = {Q (x) | Q (x) \in L^{p} (W, X), D^{α} u \in L^{p} (W, X), x \in R^{1}, α \leq k}$ ，当X为尔伯特空间时， $H^{k} (W; X) = W^{k, 2} (W; X)$ 。

接下来介绍将在干大气三维粘性原始方程组与湿大气三维粘性原始方程组中出现的算子。标量函数和向量函数在水平方向上的散度 $d i v$ 、梯度 $\nabla = g r a d$ 、协变导数 $\nabla_{\vec{v}}$ 与标量函数和向量函数的Laplace-Beltrami算子 $Δ$ 的定义分别如下

$d i v \vec{v} = d i v (v_{θ} e_{θ} + v_{ϕ} e_{ϕ}) = \frac{1}{\sin θ} (\frac{\partial (v_{θ} \sin θ)}{\partial θ} + \frac{\partial v_{ϕ}}{\partial ϕ})$ ， (15)

$\nabla T = g r a d T = \frac{\partial T}{\partial θ} e_{θ} + \frac{1}{\sin θ} \frac{\partial T}{\partial ϕ} e_{ϕ}$ ， (16)

$g r a d Φ_{s} = \frac{\partial Φ_{s}}{\partial θ} e_{θ} + \frac{1}{\sin θ} \frac{\partial Φ_{s}}{\partial ϕ} e_{ϕ}$ ，(17)

$\nabla_{\vec{v}} {\vec{v}}_{1} = (v_{θ} \frac{\partial v_{1 θ}}{\partial θ} + \frac{v_{ϕ}}{\sin θ} \frac{\partial v_{1 ϕ}}{\partial ϕ} - v_{ϕ} v_{1 ϕ} \cot θ) e_{θ} + (v_{θ} \frac{\partial v_{1 θ}}{\partial θ} + \frac{v_{ϕ}}{\sin θ} \frac{\partial v_{1 ϕ}}{\partial ϕ} + v_{ϕ} v_{1 θ} \cot θ) e_{ϕ}$ ， (18)

$\nabla_{\vec{v}} T = v_{θ} \frac{\partial T}{\partial θ} + \frac{v_{ϕ}}{\sin θ} \frac{\partial T}{\partial ϕ}$ ，(19)

$\nabla_{\vec{v}} q = v_{θ} \frac{\partial q}{\partial θ} + \frac{v_{ϕ}}{\sin θ} \frac{\partial q}{\partial ϕ}$ ，(20)

$Δ T = d i v (g r a d T) = \frac{1}{\sin θ} [\frac{\partial}{\partial θ} (\sin θ \frac{\partial T}{\partial θ}) + \frac{1}{\sin θ} \frac{\partial^{2} T}{\partial ϕ^{2}}]$ ，(21)

$Δ q = d i v (g r a d q) = \frac{1}{\sin θ} [\frac{\partial}{\partial θ} (\sin θ \frac{\partial q}{\partial θ}) + \frac{1}{\sin θ} \frac{\partial^{2} q}{\partial ϕ^{2}}]$ ，(22)

$Δ \vec{v} = (Δ v_{θ} - \frac{2 \cos θ}{\sin^{2} θ} \frac{\partial v_{ϕ}}{\partial ϕ} - \cot^{2} θ v_{θ}) e_{θ} + (Δ v_{ϕ} + \frac{2 \cos θ}{\sin^{2} θ} \frac{\partial v_{θ}}{\partial ϕ} - \cot^{2} θ v_{ϕ}) e_{ϕ}$ ，(23)

其中 $\vec{v} = v_{θ} e_{θ} + v_{ϕ} e_{ϕ}$ ， ${\vec{v}}_{1} = v_{1 θ} e_{θ} + v_{1 ϕ} e_{ϕ} \in C^{\infty} (T Ω | T S^{2})$ ， $T \in C^{\infty} (Ω)$ ， $Φ_{s} \in C^{\infty} (S^{2})$ 。此处的(23)式是修正过后向量函数的Laplace-Beltrami算子，计算过程参见文献[13]。

最后定义工作空间，令

$Θ_{1} : = {\vec{v} : \vec{v} \in C_{0}^{\infty} (T Ω | T S^{2}), {\frac{\partial \vec{v}}{\partial ξ} |}_{ξ = 0} = 0, {\frac{\partial \vec{v}}{\partial ξ} |}_{ξ = 1} = 0, \int_{0}^{1} d i v \vec{v} d ξ = 0}$ ，

$Θ_{2} : = {T : T \in C_{0}^{\infty} (Ω), {\frac{\partial T}{\partial ξ} |}_{ξ = 0} = 0, {\frac{\partial T}{\partial ξ} |}_{ξ = 1} = - α_{s} T}$ ，

$Θ_{3} : = {q : q \in C_{0}^{\infty} (Ω), {\frac{\partial q}{\partial ξ} |}_{ξ = 0} = 0, {\frac{\partial q}{\partial ξ} |}_{ξ = 1} = - β_{s} q} .$

$V_{1} = Θ_{1}$ 是关于范数 ${‖ \cdot ‖}_{1}$ 的闭包， $V_{2} = Θ_{2}$ 是关于范数 ${‖ \cdot ‖}_{1}$ 的闭包， $V_{3} = Θ_{3}$ 是关于范数 ${‖ \cdot ‖}_{1}$ 的闭包， $H_{1} = Θ_{1}$ 是关于范数 ${| \cdot |}_{2}$ 的闭包， $V = V_{1} \times V_{2} \times V_{3}$ ， $H = H_{1} \times L^{2} (Ω) \times L^{2} (Ω)$ 。

在空间V与H中的内积与范数的定义如下：

${(\vec{v}, v_{1})}_{V_{1}} = \int_{Ω} (\nabla_{e_{θ}} \vec{v} \cdot \nabla_{e_{θ}} {\vec{v}}_{1} + \nabla_{e_{ϕ}} \vec{v} \cdot \nabla_{e_{ϕ}} {\vec{v}}_{1} + \frac{\partial \vec{v}}{\partial ξ} \cdot \frac{\partial {\vec{v}}_{1}}{\partial ξ})$ ， $‖ \vec{v} ‖ = {(\vec{v}, \vec{v})}_{V_{1}}^{1 / 2}, \forall \vec{v}, {\vec{v}}_{1} \in V_{1}$ ，

${(T, T_{1})}_{V_{2}} = \int_{Ω} (g r a d T \cdot g r a d T_{1} + \frac{\partial T}{\partial ξ} \cdot \frac{\partial T_{1}}{\partial ξ})$ ， $‖ T ‖ = {(T, T)}_{V_{2}}^{1 / 2}, \forall T, T_{1} \in V_{2}$ ，

${(q, q_{1})}_{V_{3}} = \int_{Ω} (g r a d q \cdot g r a d q_{1} + \frac{\partial q}{\partial ξ} \cdot \frac{\partial q_{1}}{\partial ξ})$ ， $‖ q ‖ = {(q, q)}_{V_{3}}^{1 / 2}, \forall q, q_{1} \in V_{3}$ ，

${(U, U_{1})}_{H} = (v_{θ}, {(v_{1})}_{θ}) + (v_{ϕ}, {(v_{1})}_{ϕ}) + (T, T_{1}) + (q, q_{1})$ ， ${(U, U_{1})}_{V} = {(\vec{v}, {\vec{v}}_{1})}_{V_{1}} + {(T, T_{1})}_{V_{2}} + {(q, q_{1})}_{V_{3}}$ ， $‖ U ‖ = {(U, U)}_{V}^{1 / 2}$ ， ${| U |}_{2} = {(U, U)}_{H}^{1 / 2}$ ， $\forall U = (\vec{v}, T, q), U_{1} = ({\vec{v}}_{1}, T_{1}, q_{1}) \in V$ ，这里 $(\cdot, \cdot)$ 为 $H_{1}$ 、 $L^{2} (Ω)$ 中的 $L^{2}$ 内积。

2.3. 相关引理

引理2.3.1 [14]令 $H_{1}^{⊥}$ 为 $H_{1}$ 在 $L^{2} (T Ω | T S^{2})$ 中的正交补，则

$H_{1}^{⊥} = {\vec{v} \in L^{2} (T Ω | T S^{2}) | \vec{v} = g r a d l, l \in H^{1} (S^{2})}$ ，

$H_{1} = {\vec{v} \in L^{2} (T Ω | T S^{2}) | \int_{0}^{1} d i v \vec{v} d ξ = 0}$ ，

$V_{1} = {\vec{v} \in H_{0}^{1} (T Ω | T S^{2}) | \int_{0}^{1} d i v \vec{v} d ξ = 0} .$

引理2.3.2令 $\vec{v} = (v_{θ}, v_{ϕ})$ ， ${\vec{v}}_{1} = (v_{1}_{θ}, v_{1}_{ϕ}) \in C^{\infty} (T Ω | T S^{2})$ ， $p \in C^{\infty} (S^{2})$ ，则有

$\int_{S^{2}} p d i v \vec{v} = - \int_{S^{2}} \nabla p \cdot \vec{v}$ ， (24)

特别的，

$\int_{S^{2}} d i v \vec{v} = 0$ ， (25)

$\int_{Ω} (- Δ \vec{v}) \cdot {\vec{v}}_{1} = \int_{Ω} (\nabla_{e_{θ}} \vec{v} \cdot \nabla_{e_{θ}} {\vec{v}}_{1} + \nabla_{e_{ϕ}} \vec{v} \cdot \nabla_{e_{ϕ}} {\vec{v}}_{1}) .$ (26)

证明利用(15)，(16)与Stokes定理，就可得到(24)(可参考文献[15] [16])。又由(18)至(23)，通过计算即可得到(26)。

引理2.3.3 [4]对任意 $h \in C^{\infty} (S^{2})$ ， $\vec{v} \in C^{\infty} (T Ω | T S^{2})$ ，有

$\int_{S^{2}} \nabla_{\vec{v}} h + \int_{S^{2}} h d i v \vec{v} = \int_{S^{2}} d i v (h \vec{v}) = 0.$

引理2.3.4 [14]令 $\vec{v}, {\vec{v}}_{1} \in V_{1}, T \in V_{2}$ ，则有

$\int_{Ω} ((\nabla_{\vec{v}} {\vec{v}}_{1} + (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) \frac{\partial {\vec{v}}_{1}}{\partial ξ}) {\vec{v}}_{1}) d Ω = 0$ ，(27)

$\int_{Ω} ((\nabla_{\vec{v}} T + (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) \frac{\partial T}{\partial ξ}) T) d Ω = 0$ ，(28)

$\int_{Ω} ((\nabla_{\vec{v}} q + (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) \frac{\partial q}{\partial ξ}) q) d Ω = 0$ ，(29)

$\int_{Ω} (\int_{ξ}^{1} \frac{b P}{p} g r a d ((1 + a q) T) d ξ^{'} \cdot \vec{v} - \frac{b P}{p} (1 + a q) (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) \cdot T) d Ω = 0.$ (30)

3. 主要结论及其证明

3.1. 近似解的存在性

现在设 $w_{j} (j = 1, 2, \dots)$ 是 $L^{2} (Ω)$ 中由斯托克斯算子A的特征函数组成的完全正交基，并定义方程组(8)~(14)近似解 $U_{n} = ({\vec{v}}_{n}, T_{n}, q_{n})$ 的结构如下

${\vec{v}}_{n} = (\sum_{j = 1}^{n} a_{j n} (t) w_{j}, \sum_{j = 1}^{n} b_{j n} (t) w_{j}) = (v_{n θ}, v_{n ϕ}) = v_{n θ} e_{θ} + v_{n ϕ} e_{ϕ}$ ，

$T_{n} = \sum_{j = 1}^{n} c_{k n} (t) w_{j}$ ， $q_{n} = \sum_{j = 1}^{n} c_{j n} (t) w_{j} .$

考虑如下非线性微分方程组

$\begin{array}{l} (\frac{\partial {\vec{v}}_{n}}{\partial t} + \nabla_{{\vec{v}}_{n}} {\vec{v}}_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial {\vec{v}}_{n}}{\partial ξ} + \frac{f}{R_{0}} k \times {\vec{v}}_{n} + g r a d Φ_{s} \\ + \int_{ξ}^{1} \frac{b P}{p} g r a d [(1 + a q_{n}) T_{n}] d ξ^{'} - \frac{1}{{Re}_{1}} Δ {\vec{v}}_{n} - \frac{1}{{Re}_{2}} \frac{\partial^{2} {\vec{v}}_{n}}{\partial ξ^{2}}, w_{j}) = 0, \end{array}$ (31)

$(\frac{\partial T_{n}}{\partial t} + \nabla_{{\vec{v}}_{n}} T_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ} - \frac{b P}{p} (1 + a q_{n}) (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) - \frac{1}{R t_{1}} Δ T_{n} - \frac{1}{R t_{2}} \frac{\partial^{2} T_{n}}{\partial ξ^{2}}, w_{j}) = (Q_{1}, w_{j}),$ (32)

$(\frac{\partial q_{n}}{\partial t} + \nabla_{{\vec{v}}_{n}} q_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial q_{n}}{\partial ξ} - \frac{1}{R q_{1}} Δ q_{n} - \frac{1}{R q_{2}} \frac{\partial^{2} q_{n}}{\partial ξ^{2}}, w_{j}) = (Q_{2}, w_{j}),$ (33)

$\begin{matrix} U (t + W) = (\vec{v} (t + W), T (t + W), q (t + W)) \\ = (\vec{v}, T, q) = U (t) x \in Ω, t \in R^{1} . \end{matrix}$ (34)

设 ${W^{'}}_{n}$ 是 $L^{2} (Ω)$ 的子空间，由 $w_{1}, w_{2}, \dots, w_{n}$ 组成。易知任意 $V_{n} = ({\vec{u}}_{n}, C_{n}, S_{n}) \in C^{1} (W, {W^{'}}_{n})$ 对于线性方程组

$\begin{array}{l} ({\vec{v}}_{n t} + \frac{f}{R_{0}} k \times {\vec{v}}_{n} + g r a d Φ_{s} - \frac{1}{{Re}_{1}} Δ {\vec{v}}_{n} - \frac{1}{{Re}_{2}} \frac{\partial^{2} {\vec{v}}_{n}}{\partial ξ^{2}}, w_{j}) \\ = (- \nabla_{{\vec{u}}_{n}} {\vec{u}}_{n} - (\int_{ξ}^{1} d i v {\vec{u}}_{n} d ξ^{'}) \frac{\partial {\vec{u}}_{n}}{\partial ξ} - \int_{ξ}^{1} \frac{b P}{p} g r a d [(1 + S_{n}) C_{n}] d ξ^{'}, w_{j}), \end{array}$

$(T_{n t} - \frac{1}{R t_{1}} Δ T_{n} - \frac{1}{R t_{2}} \frac{\partial^{2} T_{n}}{\partial ξ^{2}}, w_{j}) = (- \nabla_{{\vec{u}}_{n}} C_{n} - (\int_{ξ}^{1} d i v {\vec{u}}_{n} d ξ^{'}) \frac{\partial C_{n}}{\partial ξ} + \frac{b P}{p} (1 + a S_{n}) (\int_{ξ}^{1} d i v {\vec{u}}_{n} d ξ^{'}) + Q_{1}, w_{j}),$

$(q_{n t} - \frac{1}{R q_{1}} Δ q_{n} - \frac{1}{R q_{2}} \frac{\partial^{2} q_{n}}{\partial ξ^{2}}, w_{j}) = (- \nabla_{{\vec{u}}_{n}} S_{n} - (\int_{ξ}^{1} d i v {\vec{u}}_{n} d ξ^{'}) \frac{\partial S_{n}}{\partial ξ} + Q_{2}, w_{j})$

存在时间周期解 $U_{n} = ({\vec{v}}_{n}, T_{n}, q_{n}) \in C^{1} (W, {W^{'}}_{n})$ 且唯一。所以构造在 $C^{1} (W, {W^{'}}_{n})$ 中连续且紧的函数 $F : V_{n} \to U_{n}$ ，然后运用Leray-schauder不动点定理就可以得到(31)~(34)近似解的存在性。应用不动点定理，只需证明用

$δ (\nabla_{{\vec{v}}_{n}} {\vec{v}}_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial {\vec{v}}_{n}}{\partial ξ} + \int_{ξ}^{1} \frac{b P}{p} g r a d [(1 + a q_{n}) T_{n}] d ξ^{'})$ ，

$δ (\nabla_{{\overset{⇀}{v}}_{n}} T_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ} - \frac{b P}{p} (1 + a q_{n}) (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}))$ ，

$δ (\nabla_{{\overset{⇀}{v}}_{n}} q_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial q_{n}}{\partial ξ})$

( $0 < δ < 1$ )替换非线性项

$\nabla_{{\vec{v}}_{n}} {\vec{v}}_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial {\vec{v}}_{n}}{\partial ξ} + \int_{ξ}^{1} \frac{b P}{p} g r a d T_{n} d ξ^{'}$ ，

$\nabla_{{\overset{⇀}{v}}_{n}} T_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ} - \frac{b P}{p} (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'})$ ，

$\nabla_{{\overset{⇀}{v}}_{n}} q_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial q_{n}}{\partial ξ}$

后(31)~(34)的所有可能解的有界性，即证明

$\sup_{0 \leq t \leq W} ({| {\vec{v}}_{n} (t) |}_{2}^{2} + {| T_{n} (t) |}_{2}^{2} + {| q_{n} (t) |}_{2}^{2}) \leq C$ ，

其中C是一个与 $δ$ 无关的常数。

用 $a_{j n} (t), b_{j n} (t) (j = 1, 2, \dots, n)$ 分别乘以方程(31)，并对 $j = 1, 2, \dots, n$ 求和可以得到两个方程，再将这两个方程相加就可得到

$\begin{array}{l} ({\vec{v}}_{n t} + \frac{f}{R_{0}} k \times {\vec{v}}_{n} + g r a d Φ_{s} - \frac{1}{{Re}_{1}} Δ {\vec{v}}_{n} - \frac{1}{{Re}_{2}} \frac{\partial^{2} {\vec{v}}_{n}}{\partial ξ^{2}}, {\vec{v}}_{n}) \\ = (- δ \nabla_{{\vec{v}}_{n}} {\vec{v}}_{n} - δ (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial {\vec{v}}_{n}}{\partial ξ} - δ \int_{ξ}^{1} \frac{b P}{p} g r a d [(1 + a q_{n}) T_{n}] d ξ^{'}, {\vec{v}}_{n}) . \end{array}$ (35)

对(35)运用分部积分，利用引理2.2.1，引理2.3.2和引理2.3.4，有

$\frac{1}{2} \frac{d}{d t} {| {\vec{v}}_{n} |}_{2}^{2} + \frac{1}{R e_{1}} \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + \frac{1}{R e_{2}} \int_{Ω} {| \frac{\partial {\vec{v}}_{n}}{\partial ξ} |}^{2} = - δ \int_{Ω} (\int_{ξ}^{1} \frac{b P}{p} [g r a d (1 + a q_{n}) T_{n}] d ξ^{'}) {\vec{v}}_{n} .$ (36)

用 $c_{j n} (t) (j = 1, 2, \dots, n)$ 乘以方程(32)，并对 $j = 1, 2, \dots, n$ 求和，再对其应用分部积分和引理2.3.4可以得到

$\frac{1}{2} \frac{d}{d t} {| T_{n} |}_{2}^{2} + \frac{1}{R t_{1}} {\int_{Ω} | \nabla T_{n} |}^{2} + \frac{1}{R t_{2}} {\int_{Ω} | \frac{\partial T_{n}}{\partial ξ} |}^{2} + \frac{α_{s}}{R t_{2}} {| {T_{n} |}_{ξ = 1} |}_{2}^{2} = δ \int_{Ω} \frac{b P}{p} (1 + a q_{n}) T_{n} (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) + \int_{Ω} Q_{1} T_{n} .$ (37)

类似的用 $d_{j n} (t) (j = 1, 2, \dots, n)$ 乘以方程(33)，可以得到

$\frac{1}{2} \frac{d}{d t} {| q_{n} |}_{2}^{2} + \frac{1}{R q_{1}} {\int_{Ω} | \nabla q_{n} |}^{2} + \frac{1}{R q_{2}} {\int_{Ω} | \frac{\partial q_{n}}{\partial ξ} |}^{2} + \frac{β_{s}}{R t_{2}} {| {q_{n} |}_{ξ = 1} |}_{2}^{2} = \int_{Ω} Q_{2} q_{n} .$ (38)

所以由(36)~(38)和引理2.3.4，可得

$\begin{array}{l} \frac{1}{2} \frac{d}{d t} ({| {\vec{v}}_{n} |}_{2}^{2} + {| T_{n} |}_{2}^{2} + {| q_{n} |}_{2}^{2}) + \frac{1}{{Re}_{1}} \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + \frac{1}{{Re}_{2}} \int_{Ω} {| \frac{\partial {\vec{v}}_{n}}{\partial ξ} |}^{2} + \frac{1}{R t_{1}} \int_{Ω} {| \nabla T_{n} |}^{2} + \frac{1}{R t_{2}} \int_{Ω} {| \frac{\partial T_{n}}{\partial ξ} |}^{2} \\ + \frac{α_{s}}{R t_{2}} {| {T_{n} |}_{ξ = 1} |}_{2}^{2} + \frac{1}{R q_{1}} {\int_{Ω} | \nabla q_{n} |}^{2} + \frac{1}{R q_{2}} {\int_{Ω} | \frac{\partial q_{n}}{\partial ξ} |}^{2} + \frac{β_{s}}{R q_{2}} {| {q_{n} |}_{ξ = 1} |}_{2}^{2} = \int_{Ω} Q_{1} T_{n} + \int_{Ω} Q_{2} q_{n} . \end{array}$ (39)

由Young不等式，可知

$| \int_{Ω} Q_{1} T_{n} | \leq c {| Q_{1} |}_{2}^{2} + ε {| T_{n} |}_{2}^{2}$ ， $| \int_{Ω} Q_{2} q_{n} | \leq c {| Q_{2} |}_{2}^{2} + ε {| q_{n} |}_{2}^{2} .$ (40)

又有

$\int_{Ω} T_{n} {(t, θ, ϕ, ξ)}^{2} = \int_{Ω} {(- \int_{ξ}^{1} \frac{\partial T_{n}}{\partial ξ^{'}} d ξ^{'} + {T_{n} |}_{ξ = 1})}^{2} \leq 2 {| \frac{\partial T_{n}}{\partial ξ} |}_{2}^{2} + 2 {| {T_{n} |}_{ξ = 1} |}_{2}^{2}$ ，(41)

用一样的方法可得

$\int_{Ω} q_{n} {(t, θ, ϕ, ξ)}^{2} \leq 2 {| \frac{\partial q_{n}}{\partial ξ} |}_{2}^{2} + 2 {| {q_{n} |}_{ξ = 1} |}_{2}^{2} .$ (42)

因此由(39)~(42)，有下面不等式成立

$\frac{d}{d t} ({| {\vec{v}}_{n} |}_{2}^{2} + {| T_{n} |}_{2}^{2} + {| q_{n} |}_{2}^{2}) + c_{1} {‖ {\vec{v}}_{n} ‖}^{2} + c_{2} {‖ T_{n} ‖}^{2} + c_{2} {‖ q_{n} ‖}^{2} + \frac{2 α_{s}}{R t_{2}} {| {T_{n} |}_{ξ = 1} |}_{2}^{2} + \frac{2 β_{s}}{R q_{2}} {| {q_{n} |}_{ξ = 1} |}_{2}^{2} \leq c K^{2} + c {K^{'}}^{2},$ (43)

其中 $c_{1} = \min {\frac{2}{{Re}_{1}}, \frac{2}{{Re}_{2}}}$ ， $c_{2} = \min {\frac{2}{R t_{1}}, \frac{2 - 2 ε}{R t_{2}}}$ ， $c_{2} = \min {\frac{2}{R q_{1}}, \frac{2 - 2 ε}{R q_{2}}}$ ， $ε$ 是一个足够小的正的常数.

然后利用 ${\vec{v}}_{n}, T_{n}$ 的周期性，可以得到

$\int_{0}^{W} ({‖ {\vec{v}}_{n} ‖}^{2} + {‖ T_{n} ‖}^{2} + {| {T_{n} |}_{ξ = 1} |}_{2}^{2} + {‖ q_{n} ‖}^{2} + {| {q_{n} |}_{ξ = 1} |}_{2}^{2}) \leq c W (K^{2} + {K^{'}}^{2})$ ，(44)

$\sup_{0 \leq t \leq W} ({| {\vec{v}}_{n} (t) |}_{2}^{2} + {| T_{n} (t) |}_{2}^{2} + {| q_{n} (t) |}_{2}^{2}) \leq c (μ_{1}^{- 1} + μ_{2}^{- 1} + μ_{3}^{- 1} + 2 W) (K^{2} + {K^{'}}^{2}) = E_{1} .$ (45)

这里 $E_{1}$ 独立于 $δ$ 和n (利用poincare不等式可以有 ${| q_{n} |}_{2}^{2} \leq μ_{3}^{- 1} {‖ q_{n} ‖}_{}^{2}$ )。至此证明了 $U_{n} = ({\vec{v}}_{n}, T_{n}, q_{n}) \in C^{1} (W, {W^{'}}_{n})$ 为方程组(31)~(34)的近似解。

3.2. 先验估计

引理3.2.1 设 $({\vec{v}}_{n} (t), T_{n} (t), q_{n} (t))$ 是满足上述条件的方程组(31)~(34)的近似解，令 $K_{1} \equiv \int_{0}^{W} {| Q_{1} |}_{3}^{3} d t$ ， $K_{2} \equiv \int_{0}^{W} {| Q_{1} |}_{4}^{4} d t$ ， ${K^{'}}_{2} \equiv \int_{0}^{W} {| Q_{2} |}_{4}^{4} d t$ ，则有

$\sup_{t} {| q_{n} |}_{4}^{4} \leq c (μ_{3}^{- 1} W) + 2 c K^{'} = E_{2}$ ，

$\sup_{t} {| T_{n} |}_{3}^{3} \leq (μ_{2}^{- 1} + 2 W) C (E_{1}, E_{2}, K, K^{'}, K_{1}) = E_{3}$ ，

$\sup_{t} {| T_{n} |}_{4}^{4} \leq (μ_{2}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{3}, K, K^{'}, K_{2}) = E_{4}$ ，

$\sup_{t} {| {\vec{v}}_{n} |}_{3}^{3} \leq (μ_{1}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{4}, K, K^{'}) = E_{5}$ ，

$\sup_{t} {| {\vec{v}}_{n} |}_{4}^{4} \leq (μ_{1}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{4}, E_{5}, K, K^{'}) = E_{6}$ ，

其中 $E_{i} (i = 2, 3, 4, 5, 6)$ 是与n无关的。

证明用 $d_{j n} (t) (j = 1, 2, \dots, n)$ 乘以方程(33)，并对 $j = 1, 2, \dots, n$ 求和有

$(q_{n t} - \frac{1}{R q_{1}} Δ q_{n} - \frac{1}{R q_{2}} \frac{\partial^{2} q_{n}}{\partial ξ^{2}}, q_{n}) = (- \nabla_{{\vec{v}}_{n}} q_{n} - (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial q_{n}}{\partial ξ} + Q_{2}, q_{n})$ ，(46)

接着用 ${| q_{n} |}^{2}$ 乘以(46)，并使用分部积分有

$\begin{array}{l} \frac{1}{4} \frac{d {| q_{n} |}_{4}^{4}}{d t} + \frac{3}{R q_{1}} \int_{Ω} {| \nabla q_{n} |}^{2} {| q_{n} |}^{2} + \frac{3}{R q_{2}} \int_{Ω} {| \frac{\partial q_{n}}{\partial ξ} |}^{2} {| q_{n} |}^{2} + \frac{β_{s}}{R q_{2}} \int_{Ω} {| {q_{n} |}_{ξ = 1} |}^{4} \\ = - \int_{Ω} (\nabla_{{\vec{v}}_{n}} q_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial q_{n}}{\partial ξ}) {| q_{n} |}^{q} q_{n} + \int_{Ω} Q_{2} {| q_{n} |}^{2} q_{n} . \end{array}$ (47)

然后对(47)的右边两项进行估计。运用分部积分与引理2.3.3，可以得到

$\begin{array}{l} \int_{Ω} (\nabla_{{\vec{v}}_{n}} q_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial q_{n}}{\partial ξ}) {| q_{n} |}^{2} q_{n} \\ = \frac{1}{4} \int_{Ω} \nabla_{{\vec{v}}_{n}} q_{n}^{4} + \int_{S^{2}} [\int_{0}^{1} (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) d (\frac{1}{4} q_{n}^{4})] = \frac{1}{4} \int_{Ω} (\nabla_{{\vec{v}}_{n}} q_{n}^{4} + q_{n}^{4} d i v {\vec{v}}_{n}) = 0. \end{array}$ (48)

利用Young不等式，有

$\int_{Ω} Q_{2} {| q_{n} |}^{2} q_{n} \leq c {| Q_{2} |}_{4}^{4} + ε {‖ q_{n} ‖}_{4}^{4}$ ，(49)

又由 $q_{n}^{4} (θ, ϕ, ξ) = - \int_{ξ}^{1} \frac{\partial q_{n}^{4}}{\partial ξ^{'}} d ξ^{'} + {q_{n}^{4} |}_{ξ = 1}$ ，可得

${| q_{n} |}_{4}^{4} \leq 4 \int_{S^{2}} (\int_{0}^{1} (\int_{ξ}^{1} {| q_{n} |}^{3} | \frac{\partial q_{n}}{\partial ξ^{'}} |)) + {| {q_{n} |}_{ξ = 1} |}_{4}^{4} \leq c (\int_{Ω} {| q_{n} |}^{2} {| q_{n ξ} |}^{2}) + \frac{1}{2} \int_{Ω} q_{n}^{4} + {| {q_{n} |}_{ξ = 1} |}_{4}^{4} .$ (50)

所以由(47)~(50)有

$\frac{1}{4} \frac{d {| q_{n} |}_{4}^{4}}{d t} + \frac{3}{R q_{1}} \int_{Ω} {| \nabla q_{n} |}^{2} {| q_{n} |}^{2} + \frac{3}{R q_{2}} \int_{Ω} {| \frac{\partial q_{n}}{\partial ξ} |}^{2} {| q_{n} |}^{2} + \frac{β_{s}}{R q_{2}} \int_{Ω} {| {q_{n} |}_{ξ = 1} |}^{4} \leq c {| Q_{2} |}_{4}^{4}$ .(51)

利用 $q_{n}$ 的周期性，poincare不等式有

$\int_{0}^{W} (\frac{3}{R q_{1}} \int_{Ω} {| \nabla q_{n} |}^{2} {| q_{n} |}^{2} + \frac{3}{R q_{2}} \int_{Ω} {| \frac{\partial q_{n}}{\partial ξ} |}^{2} {| q_{n} |}^{2} + \frac{β_{s}}{R q_{2}} \int_{Ω} {| {q_{n} |}_{ξ = 1} |}^{4}) \leq c {K^{'}}_{2}$ ， (52)

$\sup_{t} {| q_{n} |}_{4}^{4} \leq c (μ_{3}^{- 1} W) + 2 c K^{'} = E_{2} .$ (53)

用 $c_{j n} (t) (j = 1, 2, \dots, n)$ 乘以方程(32)，并对 $j = 1, 2, \dots, n$ 求和有

$(T_{n t} - \frac{1}{R t_{1}} Δ T_{n} - \frac{1}{R t_{2}} \frac{\partial^{2} T_{n}}{\partial ξ^{2}}, T_{n}) = (- \nabla_{{\vec{v}}_{n}} T_{n} - (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ} + \frac{b P}{p} (1 + a q_{n}) (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) + Q_{1}, T_{n})$ ，(54)

然后用 $| T_{n} |$ 乘以(54)，并应用分部积分有

$\begin{array}{l} \frac{1}{3} \frac{d {| T_{n} |}_{3}^{3}}{d t} + \frac{2}{R t_{1}} \int_{Ω} {| \nabla T_{n} |}^{2} | T_{n} | + \frac{2}{R t_{2}} \int_{Ω} {| \frac{\partial T_{n}}{\partial ξ} |}^{2} | T_{n} | + \frac{α_{s}}{R t_{2}} \int_{Ω} {| {T_{n} |}_{ξ = 1} |}^{3} \\ = - \int_{Ω} (\nabla_{{\vec{v}}_{n}} T_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ}) | T_{n} | T_{n} + \int_{Ω} \frac{b P}{p} (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) | T_{n} | T_{n} + \int_{Ω} Q_{1} | T_{n} | T_{n} . \end{array}$ (55)

接着对(55)的右边三项进行估计。与得到式子(4.16)方法类似，可以得到

$\int_{Ω} (\nabla_{{\vec{v}}_{n}} T_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ}) | T_{n} | T_{n} = 0.$ (56)

利用Gagliardo-Nirenberg不等式，Young不等式与Hölder不等式，可以发现

$\int_{Ω} \frac{b P}{p} (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) | T_{n} | T_{n} \leq c {| d i v {\vec{v}}_{n} |}_{2}^{2} + c {| T_{n} |}_{2}^{2} {‖ T_{n} ‖}^{2} \leq c \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + c {| T_{n} |}_{2}^{2} {‖ T_{n} ‖}^{2},$ (57)

$\begin{matrix} \int_{Ω} \frac{a b P}{p} q_{n} (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) | T_{n} | T_{n} \leq c \int_{0}^{1} {{(\int_{S^{2}} q_{n}^{4})}^{\frac{1}{4}} {[\int_{S^{2}} {({| T_{n} |}^{\frac{3}{2}})}^{\frac{16}{3}}]}^{\frac{1}{4}}} {| d i v {\vec{v}}_{n} |}_{2} \\ \leq c {| q_{n} |}_{4} {| T_{n} |}_{3}^{\frac{3}{4}} {(\int_{Ω} | T_{n} | {| \nabla T_{n} |}^{2})}^{\frac{5}{12}} {| d i v {\vec{v}}_{n} |}_{2} \\ \leq c {| q_{n} |}_{4}^{24} + c {| T_{n} |}_{2}^{6} + c {‖ T_{n} ‖}^{2} + c \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + ε \int_{Ω} | T_{n} | {| \nabla T_{n} |}^{2}, \end{matrix}$ (58)

$| \int_{Ω} Q_{1} | T_{n} | T_{n} | \leq {| Q_{1} |}_{3} {| T_{n} |}_{3}^{2} \leq c {| Q_{1} |}_{3}^{3} + c {| T_{n} |}_{2}^{6} + c {‖ T_{n} ‖}^{2} .$ (59)

所以由(55)~(59)有

$\begin{array}{l} \frac{1}{3} \frac{d {| T_{n} |}_{3}^{3}}{d t} + \frac{2}{R t_{1}} \int_{Ω} {| \nabla T_{n} |}^{2} | T_{n} | + \frac{2}{R t_{2}} \int_{Ω} {| \frac{\partial T_{n}}{\partial ξ} |}^{2} | T_{n} | + \frac{α_{s}}{R t_{2}} \int_{Ω} {| {T_{n} |}_{ξ = 1} |}^{3} \\ \leq 2 c \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + c ({| T_{n} |}_{2}^{2} + 2) {‖ T_{n} ‖}^{2} + c {| q_{n} |}_{4}^{24} + 2 c {| T_{n} |}_{2}^{6} + c {| Q_{1} |}_{3}^{3} . \end{array}$ (60)

利用 $T_{n}$ 的周期性(44)、(45)和(53)有，

$\int_{0}^{W} (\frac{2}{R t_{1}} {\int_{Ω} | \nabla T_{n} |}^{2} | T_{n} | + \frac{2}{R t_{2}} \int_{Ω} {| \frac{\partial T_{n}}{\partial ξ} |}^{2} | T_{n} | + \frac{α_{s}}{R t_{2}} \int_{Ω} {| {T_{n} |}_{ξ = 1} |}^{3}) \leq C (E_{1}, E_{2}, K, K^{'}, K_{1}) W$ ， (61)

$\sup_{t} {| T_{n} |}_{3}^{3} \leq (μ_{2}^{- 1} + 2 W) C (E_{1}, E_{2}, K, K^{'}, K_{1}) = E_{3} .$ (62)

同样用 ${| T_{n} |}^{2}$ 乘以(54)，并应用分部积分有

$\begin{array}{l} \frac{1}{4} \frac{d {| T_{n} |}_{4}^{4}}{d t} + \frac{3}{R t_{1}} {\int_{Ω} | \nabla T_{n} |}^{2} {| T_{n} |}^{2} + \frac{3}{R t_{2}} \int_{Ω} {| \frac{\partial T_{n}}{\partial ξ} |}^{2} {| T_{n} |}^{2} + \frac{α_{s}}{R t_{2}} \int_{Ω} {| {T_{n} |}_{ξ = 1} |}^{4} \\ = - \int_{Ω} (\nabla_{{\vec{v}}_{n}} T_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ}) {| T_{n} |}^{2} T_{n} + \int_{Ω} Q_{1} {| T_{n} |}^{2} T_{n} + \int_{Ω} \frac{b P}{p} (1 + a q_{n}) (\int_{ξ}^{1} d i v \vec{v} d ξ^{'}) {| T_{n} |}^{2} T_{n} . \end{array}$ (63)

与得到式子(4.16)的方法类似，也可以得到

$\int_{Ω} (\nabla_{{\vec{v}}_{n}} T_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial T_{n}}{\partial ξ}) {| T_{n} |}^{2} T_{n} = 0$ ，(64)

再利用Gagliardo-Nirenberg不等式、Young不等式和Hölder不等式，有

$\begin{matrix} \int_{Ω} \frac{b P}{p} (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) {| T_{n} |}^{2} T_{n} \leq c {| d i v {\vec{v}}_{n} |}_{2}^{2} + c {| T_{n} |}_{6}^{6} \\ \leq c \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + c {| T_{n} |}_{2}^{4} {‖ T_{n} ‖}^{2}, \end{matrix}$ (65)

$\begin{matrix} \int_{Ω} \frac{a b P}{p} q_{n} (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) {| T_{n} |}^{2} T_{n} \leq c \int_{0}^{1} {{(\int_{S^{2}} q_{n}^{4})}^{\frac{1}{4}} {[\int_{S^{2}} {({| T_{n} |}^{2})}^{6}]}^{\frac{1}{4}}} {| d i v {\vec{v}}_{n} |}_{2} \\ \leq c {| q_{n} |}_{4}^{8} + c {| T_{n} |}_{3}^{2} {‖ T_{n} ‖}^{2} + c \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + ε \int_{Ω} | T_{n} | {| \nabla T_{n} |}^{2}, \end{matrix}$ (66)

$| \int_{Ω} Q_{1} {| T_{n} |}^{2} T_{n} | \leq {| Q_{1} |}_{4} {| T_{n} |}_{4}^{3} \leq c {| Q_{1} |}_{4}^{4} + c {| T_{n} |}_{3}^{2} {‖ T_{n} ‖}^{2} .$ (67)

所以由(63)~(67)可得

$\begin{array}{l} \frac{1}{4} \frac{d {| T_{n} |}_{4}^{4}}{d t} + \frac{3}{R t_{1}} \int_{Ω} {| \nabla T_{n} |}^{2} {| T_{n} |}^{2} + \frac{3}{R t_{2}} \int_{Ω} {| \frac{\partial T_{n}}{\partial ξ} |}^{2} {| T_{n} |}^{2} + \frac{α_{s}}{R t_{2}} \int_{Ω} {| {T_{n} |}_{ξ = 1} |}^{4} \\ \leq 2 c \int_{Ω} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) + c (2 {| T_{n} |}_{3}^{2} + {| T_{n} |}_{2}^{4}) {‖ T_{n} ‖}^{2} + c {| q_{n} |}_{4}^{8} + c {| Q_{1} |}_{4}^{4} . \end{array}$ (68)

接着利用 $T_{n}$ 的周期性，式子(44)、(45)、(53)和(62)有

$\int_{0}^{W} (\frac{3}{R t_{1}} \int_{Ω} {| \nabla T_{n} |}^{2} {| T_{n} |}^{2} + \frac{3}{R t_{2}} \int_{Ω} {| \frac{\partial T_{n}}{\partial ξ} |}^{2} {| T_{n} |}^{2} + \frac{α_{s}}{R t_{2}} \int_{Ω} {| {T_{n} |}_{ξ = 1} |}^{4}) \leq C (E_{1}, E_{2}, E_{3}, K, K^{'}, K_{2}) W,$ (69)

$\sup_{t} {| T_{n} |}_{4}^{4} \leq (μ_{2}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{3}, K, K^{'}, K_{2}) = E_{4} .$ (70)

用 $a_{j n} (t), b_{j n} (t) (j = 1, 2, \dots, n)$ 分别乘以方程(31)，并对 $k = 1, 2, \dots, n$ 求和可以得到两个方程，再将这两个方程相加就可得到

$\begin{array}{l} ({\vec{v}}_{n t} + \frac{f}{R_{0}} k \times {\vec{v}}_{n} + g r a d Φ_{s} - \frac{1}{{Re}_{1}} Δ {\vec{v}}_{n} - \frac{1}{{Re}_{2}} \frac{\partial^{2} {\vec{v}}_{n}}{\partial ξ^{2}}, {\vec{v}}_{n}) \\ = (- \nabla_{{\vec{v}}_{n}} {\vec{v}}_{n} - (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial {\vec{v}}_{n}}{\partial ξ} - \int_{ξ}^{1} \frac{b P}{p} g r a d (1 + a q_{n}) T_{n} d ξ^{'}, {\vec{v}}_{n}) . \end{array}$ (71)

除此之外，用 $| {\vec{v}}_{n} |$ 乘以(71)，并运用分部积分可以得到

$\begin{array}{l} \frac{1}{3} \frac{d {| {\vec{v}}_{n} |}_{3}^{3}}{d t} + \frac{1}{{Re}_{1}} \int_{Ω} [({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) | {\vec{v}}_{n} | + \frac{4}{9} {| \nabla_{e_{θ}} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2} + \frac{4}{9} {| \nabla_{e_{ϕ}} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2}] + \frac{1}{{Re}_{2}} \int_{Ω} ({| {\vec{v}}_{n ξ} |}^{2} | {\vec{v}}_{n} | + \frac{4}{9} {| \partial_{ξ} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2}) \\ = - \int_{Ω} (\nabla_{{\vec{v}}_{n}} {\vec{v}}_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'}) \frac{\partial {\vec{v}}_{n}}{\partial ξ}) | {\vec{v}}_{n} | {\vec{v}}_{n} - \int_{Ω} \frac{b P}{p} (\int_{ξ}^{1} g r a d (1 + a q_{n}) T_{n} d ξ^{'}) | {\vec{v}}_{n} | {\vec{v}}_{n} . \end{array}$ (72)

然后对(72)的右边两项进行估计。利用Gagliardo-Nirenberg不等式至Hölder不等式和引理2.3.4，有

$\begin{array}{l} \frac{1}{3} \frac{d {| {\vec{v}}_{n} |}_{3}^{3}}{d t} + \frac{1}{{Re}_{1}} \int_{Ω} [({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) | {\vec{v}}_{n} | + \frac{4}{9} {| \nabla_{e_{θ}} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2} + \frac{4}{9} {| \nabla_{e_{ϕ}} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2}] + \frac{1}{{Re}_{2}} \int_{Ω} ({| {\vec{v}}_{n ξ} |}^{2} | {\vec{v}}_{n} | + \frac{4}{9} {| \partial_{ξ} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2}) \\ \leq {| q_{n} |}_{4}^{2} {| T_{n} |}_{4}^{2} {| {\vec{v}}_{n} |}_{2}^{2} {‖ q_{n} ‖}^{2} + {‖ T_{n} ‖}^{2} + {| T_{n} |}_{4}^{4} + {| {\vec{v}}_{n} |}_{2}^{2} . \end{array}$ (73)

再利用 ${\vec{v}}_{n}$ 的周期性，式子(44)、(45)、(53)和(62)可证得

$\begin{array}{l} \int_{0}^{W} [\frac{1}{{Re}_{1}} \int_{Ω} [({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) | {\vec{v}}_{n} | + \frac{4}{9} {| \nabla_{e_{θ}} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2} + \frac{4}{9} {| \nabla_{e_{ϕ}} {| {\vec{v}}_{n} |}^{\frac{3}{2}} |}^{2}] + \frac{1}{{Re}_{2}} \int_{Ω} ({| {\vec{v}}_{n ξ} |}^{2} | {\vec{v}}_{n} | + \frac{4}{9} {| \partial_{ξ} {| \vec{v} |}^{\frac{3}{2}} |}^{2})] \\ \leq C (E_{1}, E_{2}, E_{4}, K, K^{'}) W, \end{array}$ (74)

$\sup_{t} {| {\vec{v}}_{n} |}_{3}^{3} \leq (μ_{1}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{4}, K, K^{'}) = E_{5} .$ (75)

类似的用 ${| {\vec{v}}_{n} |}^{2}$ 乘以(71)，可以得到

$\begin{array}{l} \frac{1}{4} \frac{d {| {\vec{v}}_{n} |}_{4}^{4}}{d t} + \frac{1}{{Re}_{1}} \int_{Ω} [({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2}) {| {\vec{v}}_{n} |}^{2} + \frac{1}{2} {| \nabla_{e_{θ}} {| {\vec{v}}_{n} |}^{2} |}^{2} + \frac{1}{2} {| \nabla_{e_{ϕ}} {| {\vec{v}}_{n} |}^{2} |}^{2}] + \frac{1}{{Re}_{2}} \int_{Ω} ({| {\vec{v}}_{n ξ} |}^{2} {| {\vec{v}}_{n} |}^{2} + \frac{1}{2} {| \partial_{ξ} {| {\vec{v}}_{n} |}^{2} |}^{2}) \\ = - \int_{Ω} (\nabla_{{\vec{v}}_{n}} {\vec{v}}_{n} + (\int_{ξ}^{1} d i v {\vec{v}}_{n} d ξ^{'})) {| {\vec{v}}_{n} |}^{2} {\vec{v}}_{n} - \int_{Ω} \frac{b P}{p} (\int_{ξ}^{1} g r a d (1 + a q_{n}) T_{n} d ξ^{'}) {| {\vec{v}}_{n} |}^{2} {\vec{v}}_{n}, \end{array}$ (76)

再用类似的方法，可证得

$\sup_{t} {| {\vec{v}}_{n} |}_{4}^{4} \leq (μ_{1}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{4}, E_{5}, K, K^{'}) = E_{6} .$ (77)

至此完成了对引理3.2.1的证明。

用类似证明引理3.2.1的方法可得到如下定理与引理。

定理3.2.1 设 $({\vec{v}}_{n} (t), T_{n} (t), q_{n} (t))$ 是满足上述条件的方程组(31)~(34)的近似解，令 $K_{3} \equiv \int_{0}^{W} {| Q_{1 ξ} |}_{2}^{2} d t$ ， ${K^{'}}_{3} \equiv \int_{0}^{W} {| Q_{2 ξ} |}_{2}^{2} d t$ ，则有

$\sup_{t} {| {\vec{v}}_{n ξ} (t) |}_{2}^{2} \leq (μ_{1}^{- 1} + 2 W) C (E_{1}) = E_{7}$ ，

$\sup_{t} ({| T_{n ξ} |}_{2}^{2} + {| q_{n ξ} |}_{2}^{2}) \leq (μ_{2}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{3}, E_{4}, E_{6}, E_{7}, K, K^{'}, K_{2}, K_{3}, {K^{'}}_{3}) = E_{8}$ ，

其中 $E_{i} (i = 7, 8)$ 是与n无关的。

引理3.2.2 设 $({\vec{v}}_{n} (t), T_{n} (t))$ 是满足上述条件的方程组(31)~(34)的近似解，则有

$\sup_{t} {| {\vec{v}}_{n ξ} (t) |}_{3}^{3} \leq (μ_{1}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{4}, E_{6}, E_{7}, K, K^{'}) = E_{9},$

$\sup_{t} {| {\vec{v}}_{n ξ} (t) |}_{4}^{4} \leq (μ_{1}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{4}, E_{6}, E_{9}, K, K^{'}) = E_{10},$

$\sup_{t} {| q_{n ξ} (t) |}_{3}^{3} \leq (μ_{3}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{3}, E_{4}, E_{6}, E_{7}, E_{8}, K, K^{'}, K_{2}, K_{3}, {K^{'}}_{3}) = E_{11},$

$\sup_{t} {| q_{n ξ} (t) |}_{4}^{4} \leq (μ_{3}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{3}, E_{4}, E_{6}, E_{7}, E_{8}, E_{9}, E_{11}, K, K^{'}, K_{2}, K_{3}, {K^{'}}_{3}) = E_{12},$

$\sup_{t} {| T_{n} |}_{3}^{3} \leq (μ_{2}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{3}, E_{4}, E_{6}, E_{7}, E_{8}, E_{9}, E_{11}, K, K^{'}, K_{2}, K_{3}, {K^{'}}_{3}) = E_{13},$

$\sup_{t} {| T_{n ξ} (t) |}_{4}^{4} \leq (μ_{2}^{- 1} + 2 W) C (E_{1}, E_{2}, E_{3}, E_{4}, E_{6}, E_{7}, E_{8}, E_{9}, E_{10}, E_{12}, E_{13}, K, K^{'}, K_{2}, K_{3}, {K^{'}}_{3}) = E_{14},$

其中 $E_{i} (i = 9, 10, 11, 12, 13, 14)$ 与n无关的。

定理3.2.2 设 $({\vec{v}}_{n} (t), T_{n} (t), q_{n} (t))$ 是满足上述条件的方程组(31)~(34)的近似解，则有

$\sup_{t} ({| \nabla_{e_{θ}} {\vec{v}}_{n} |}^{2} + {| \nabla_{e_{ϕ}} {\vec{v}}_{n} |}^{2} + {| \nabla T_{n} |}_{2}^{2} + {| \nabla q_{n} |}_{2}^{2}) \leq (μ_{1}^{- 1} + μ_{2}^{- 1} + μ_{3}^{- 1} + 2 W) C (E_{2}, E_{4}, E_{6}, E_{10}, E_{12}, E_{14}, K, K^{'}) = E_{15},$

其中 $E_{15}$ 与n无关。

定理3.2.3 设 $({\vec{v}}_{n} (t), T_{n} (t))$ 是满足上述条件的方程组(31)~(34)的近似解，令 $K_{4} \equiv \int_{0}^{W} {| Q_{1 t} |}_{2}^{2} d t$ ， ${K^{'}}_{4} \equiv \int_{0}^{W} {| Q_{2 t} |}_{2}^{2} d t$ 则有

$\sup_{t} ({| {\vec{v}}_{n t} |}_{2}^{2} + {| T_{n t} |}_{2}^{2} + {| q_{n t} |}_{2}^{2}) \leq (μ_{1}^{- 1} + μ_{2}^{- 1} + μ_{3}^{- 1} + 2 W) C (E_{2}, E_{4}, E_{6}, E_{10}, E_{12}, E_{14}, K, K^{'}, K_{4}, {K^{'}}_{4}) .$

3.3. 时间周期解的存在性与唯一性

定理3.3.1 令 $Q_{1}, Q_{2} \in L^{\infty} (W, H^{1} (Ω)) (W > 0)$ ，则存在一个常数 $C_{0} = C_{0} (N) > 0$ $(N = 1, 2, 3, \dots)$ ，如果有

$K \equiv \sup_{0 \leq t \leq W} {‖ Q_{1} ‖}_{L^{N} (Ω)} \leq C_{0}$ ， $K^{'} \equiv \sup_{0 \leq t \leq W} {‖ Q_{2} ‖}_{L^{N} (Ω)} \leq C_{0}$ ，

则方程组(8)~(11)有周期为W的时间周期解 $(\vec{v}, T, q)$ ，且其满足

$(\vec{v}, T, q) \in L^{\infty} (W; V) \cap H^{1} (W, H) .$

证明本定理的证明与文献[13]第五部分证明完全类似，所以此处不给出详细证明过程。

定理3.3.2 满足定理4.4.1中所给条件的方程组(8)-(11)的时间周期解是唯一的。

证明本定理的证明与文献[13]第五部分证明完全类似，所以此处不给出详细证明过程。

4. 小结与展望

本文通过Leray-Schauder不动点定理证明到了湿大气原始方程组时间周期近似解 ${\vec{v}}_{n}, T_{n}, q_{n}$ 的存在性，证明到了 ${\vec{v}}_{n}, T_{n}, q_{n}$ 在 $L^{3}, L^{4}$ 中的有界性、 ${\vec{v}}_{n ξ}, T_{n ξ}, q_{n ξ}$ 在 $L^{2}, L^{3}, L^{4}$ 中的有界性、 ${\vec{v}}_{n}, T_{n}, q_{n}$ 在 $H^{1}$ 中的有界性、 ${\vec{v}}_{n t}, T_{n t}, q_{n t}$ 在 $L^{2}$ 中的有界性，最终得到了湿大气原始方程组时间周期弱解的存在性与唯一性。由于边界条件的限制，本文并未证明到强解的存在性，未来可进一步对周期强解与周期解的稳定性进行研究。

参考文献

[1]	Richardson, L.F. (1922) Weather Prediction by Numerical Press. Cambridge University Press.
[2]	Guo, B. and Huang, D. (2006) Existence of Weak Solutions and Trajectory Attractors for the Moist Atmospheric Equations in Geophysics. Journal of Mathematical Physics, 47, Article 083508. [Google Scholar] [CrossRef]
[3]	Guo, B.L. and Huang, D.W. (2007) Long-Time Dynamics for the 3-D Viscous Primitive Equations of Large-Scale Moist Atmosphere. Mathematics, 35, 5934-5943.
[4]	Huang, D. and Guo, B. (2008) On the Existence of Atmospheric Attractors. Science in China Series D: Earth Sciences, 51, 469-480. [Google Scholar] [CrossRef]
[5]	Kato, H. (1997) Existence of Periodic Solutions of the Navier-Stokes Equations. Journal of Mathematical Analysis and Applications, 208, 141-157. [Google Scholar] [CrossRef]
[6]	Cyranka, J., Mucha, P.B., Titi, E.S. and Zgliczyński, P. (2018) Stabilizing the Long-Time Behavior of the Forced Navier-Stokes and Damped Euler Systems by Large Mean Flow. Physica D: Nonlinear Phenomena, 369, 18-29. [Google Scholar] [CrossRef]
[7]	Guo, B. and Du, X. (2001) Existence of the Periodic Solution for the Weakly Damped Schrödinger-Boussinesq Equation. Journal of Mathematical Analysis and Applications, 262, 453-472. [Google Scholar] [CrossRef]
[8]	Guo, B.L. and Xie, B. (2017) Global Existence of Weak Solutions for Generalized Quantum MHD Equation. Annals of Applied Mathematics, 33, 111-129.
[9]	Guo, B.L. and Xie, B. (2018) Global Existence of Weak Solutions to the Three-Dimensional Full Compressible Quantum Equation. Annals of Applied Mathematics, 34, 1-31.
[10]	Galdi, G.P. and Kashiwabara, T. (2015) Strong Time-Periodic Solutions to the 3D Primitive Equations Subject to Arbitrary Large Forces. Nonlinearity, 30, Article 3979.
[11]	Long, Q. and Chen, J. (2018) Finite Time Blow-Up and Global Existence of Weak Solutions for Pseudo-Parabolic Equation with Exponential Nonlinearity. Journal of Applied Analysis & Computation, 8, 105-122. [Google Scholar] [CrossRef]
[12]	Guo, B. and Huang, D. (2011) Existence of the Universal Attractor for the 3-D Viscous Primitive Equations of Large-Scale Moist Atmosphere. Journal of Differential Equations, 251, 457-491. [Google Scholar] [CrossRef]
[13]	Luo, W. and Du, X. (2019) Existence of Time Periodic Solutions for the 3-D Viscous Primitive Equations of Large-Scale Dry Atmosphere. Journal of Applied Analysis & Computation, 9, 691-717. [Google Scholar] [CrossRef]
[14]	Lions, J.L., Temam, R. and Wang, S. (1992) New Formulations of the Primitive Equations of Atmosphere and Applications. Nonlinearity, 5, 237-288. [Google Scholar] [CrossRef]
[15]	Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd Edition, Springer.
[16]	Wang, S. (1992) On the 2D Model of Large-Scale Atmospheric Motion: Well-Posedness and Attractors. Nonlinear Analysis: Theory, Methods & Applications, 18, 17-60. [Google Scholar] [CrossRef]

为你推荐

友情链接