兰州大学机构库 >数学与统计学院
关于大尺度大气运动方程在对数压力坐标下原始简化方程的研究
Alternative TitleOn the primitive equations of the large-scale atmosphere in the log-pressure coordinates
尤波
Thesis Advisor钟承奎
2012-12-07
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword原始简化方程组 全局吸引子 Sobolev 嵌入理论 Navier-Stokes 方程组 对数压力坐标
Abstract在本篇博士学位论文中, 主要考虑在对数压力坐标下, 关于三维大尺度大气运动本原方程的如下初边值问题(1): \frac{\partial v}{\partial t}+(v\cdot\nabla)v+w\frac{\partial v}{\partial z}+f_0\vec{k}\times v+\nabla \Phi+L_1v=0,\frac{\partial\Phi}{\partial z}=\frac{RT}{H_s},\nabla\cdot\left(e^{-\frac{z}{H_s}}v\right)+\frac{\partial}{\partial z}\left(e^{-\frac{z}{H_s}}w\right)=0,\frac{\partial T}{\partial t}+v\cdot\nabla T+w\frac{\partial T}{\partial z}+L_2T=Q.\frac{\partial v}{\partial z}\right|_{\Gamma_u}=0.w\right|_{\Gamma_u}=0, \left.\left(\frac{1}{Rt_2}\frac{\partial T}{\partial z}+\alpha T\right)\right|_{\Gamma_u}=0.\frac{\partial v}{\partial z}\right|_{\Gamma_b}=0.w\right|_{\Gamma_b}=0, \left.\frac{\partial T}{\partial z}\right|_{\Gamma_b}=0.v\cdot\vec{n}\right|_{\Gamma_l}=0.\frac{\partial v}{\partial\vec{n}}\times\vec{n}\right|_{\Gamma_l}=0,\left.\frac{\partial T}{\partial\vec{n}}\right|_{\Gamma_l}=0,v(x,0)=v_0(x),T(x,0)=T_0(x), 在区域\Omega=M\times(-h,0) \end{align*} 上弱解的全局存在性,强解的全局存在唯一性以及在条件 \frac{9h^2}{4H_s^2}+\frac{4h}{\alpha H_s^2Rt_2}< 1,H_s^2Re_2\lambda>1 下强解全局吸引子的存在性. 这里 (x,y,z)\in \Omega, 其中:z=-H_s\log\left(\frac{p}{p_*}\right),H_s 是定常高度尺度,p_* 是定常参考压力. 并且记\ \Omega 的边界为:\partial\Omega=\Gamma_u\cup\Gamma_b\cup\Gamma_l, 其中: \Gamma_u=\{x\in\bar{\Omega}:z=0\},\Gamma_b={x\in\bar{\Omega}:z=-h},\Gamma_l=\{x=(\tilde{x},z)\in\bar{\Omega}:\tilde{x}\in \partial M, -h\leq z\leq 0}. 对简化方程组(1), 我们首先定义了水平速度v 的平均\bar{v} 以及它的波动\ \tilde{v}并给出了它们的一些性质, 然后把关于v的方程组 拆分成在三维区域上, 一个关于\bar{v} 的二维不可压\ Navier-Stokes\ 方程组和另一个不涉及压力项的关于\tilde{v}的方程组. 接下来通过对关于T,\tilde{v},\bar{v}和v的方程组做一些先验估计, 我们证明了弱解的全局存在性和强解的适定性. 同时也得到了 (V,V)- 和 (V,(H^2(\Omega))^3)-吸收集的存在性, 最后运用椭圆型方程正则性理论, 得到了(V,(H^3(\Omega))^3)- 吸收集的存在性. 利用Sobolev紧嵌入定理, 得到了(V,V)-和(V,(H^2(\Omega))^3)- 全局吸引子的存在性. 全文共分四章: 第一章, 介绍大尺度大气运动简化方程组的发展及研究情况和最新进展. 第二章, 给出了本文用到的函数空间以及一些不等式, 并重新公式化方程组(1). 第三章, 主要给出了方程组(1) 的一些先验估计. 第四章, 运用第三章给出的先验估计, 我们得到了弱解的全局存在性和强解的适定性,(V,V)- 和(V,(H^2(\Omega))^3)- 吸收集的存在性, 运用椭圆型方程的正则性理论, 我们得到了(V,(H^3(\Omega))^3)- 吸收集的存在性, 并运用 Sobolev紧嵌入定理得到了(V,V)- 和 (V,(H^2(\Omega))^3)- 全局吸引子的存在性.
Other AbstractIn this doctoral dissertation, we are concerned with the global existence of weak solutions, the well-posedness of strong solutions and the existence of global attractors of the following initial-boundary problems about the three dimensional viscous large-scale primitive equations in the log-pressure coordinates:(1) \frac{\partial v}{\partial t}+\left(v\cdot\nabla\right)v+w\frac{\partial v}{\partial z}+f_0\vec{k}\times v+\nabla \Phi+L_1v=0,\frac{\partial\Phi}{\partial z}=\frac{RT}{H_s},\nabla\cdot\left(e^{-\frac{z}{H_s}}v\right)+\frac{\partial}{\partial z}\left(e^{-\frac{z}{H_s}}w\right)=0,\frac{\partial T}{\partial t}+v\cdot\nabla T+w\frac{\partial T}{\partial z}+L_2T=Q. \frac{\partial v}{\partial z}\right|_{\Gamma_u}=0. w\right|_{\Gamma_u}=0, \left.\left(\frac{1}{Rt_2}\frac{\partial T}{\partial z}+\alpha T\right)\right|_{\Gamma_u}=0.\frac{\partial v}{\partial z}\right|_{\Gamma_b}=0. w\right|_{\Gamma_b}=0, \left.\frac{\partial T}{\partial z}\right|_{\Gamma_b}=0, v\cdot\vec{n}|_{\Gamma_l}=0.\frac{\partial v}{\partial\vec{n}}\times\vec{n}\right|_{\Gamma_l}=0.\frac{\partial T}{\partial\vec{n}}\right|_{\Gamma_l}=0,v(x,0)=v_0(x),T(x,0)=T_0(x),in the domain \Omega=M\times(-h,0)\subset\mathbb{R}^3, under the following conditions:\frac{9h^2}{4H_s^2}+\frac{4h}{\alpha H_s^2Rt_2}< 1, H_s^2Re_2\lambda>1. Here(x,y,z)\in \Omega is coordinates,where z=-H_s\log\left(\frac{p}{p_*}),H_sis a constant 'scale height' andp_* is a constant reference pressure. We denote the boundary of\Omega by \partial\Omega=\Gamma_u\cup\Gamma_b\cup\Gamma_l,where \Gamma_u=\{x\in\bar{\Omega}:z=0\},\Gamma_b=\{x\in\bar{\Omega}:z=-h\},\Gamma_l=\{x\in\bar{\Omega}:\tilde{x}\in \partial M, -h\leq z\leq 0\}. We first give the definition and properties of \bar{v} and \tilde{v} and we divide the equations with respect to v into two systems with respect to \bar{v} and \tilde{v} in the three dimensional domains, respectively. One is the incompressible Navier-Stokes equations in the two dimensional domains, the other is the equations with respect to\tilde{v} without the pressure term. Then we give some a priori estimates about T, \tilde{v}, \bar{v} and v, by which we prove the global existence of weak solutions, the well-posedness of strong solutions for the equations (1). At the same time, we obtain the existence of the (V,V)- and (V,(H^2(\Omega))^3)- absorbing sets. Finally, we have obtained the existence of (V,(H^3(\Omega))^3)- absorbing sets by the elliptic regularity theory, and the existence of ...
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Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225361
Collection数学与统计学院
Recommended Citation
GB/T 7714
尤波. 关于大尺度大气运动方程在对数压力坐标下原始简化方程的研究[D]. 兰州. 兰州大学,2012.
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