二维耗散准地转方程论文：二维耗散准地转方程解的渐近性.doc

二维耗散准地转方程论文：二维耗散准地转方程解的渐近性【中文摘要】地球物理流体动力学研究的重要对象之一是大气和海洋大尺度现象的动力学问题.理解动力系统的渐近行为是研究无穷维动力系统的主要内容,也是当代数学物理的重要问题之一.该文研究地球物理流体动力学中的大气、海洋简化方程.二维耗散准地转方程的模型最初由Constantin提出本文考虑的是二维耗散准地转方程解的L2衰减,高阶导数衰减以及渐近行为.本文的组织结构为：第一章,介绍了准地转方程的,以及给出研究问题所必需的预备知识.第二章,利用Laplacian算子的谱分解及迭代方法得到L2衰减速率和推出弱解的高阶导数上界.当00),则对足够大的t,Am(t)L2C(1+t)-2/m+2.第三章,本章主要是证明解的渐近行为,即(t)-(t)L20.我们选取试验函数(T)=-U(T)st(-)U()()d,帮助我们得到辅助估计lim supt(1+A)-4/2-A1/2(t)L22=0.二维耗散准地转方程的扰动方程是这里(x)和f(x,t)分别是初值扰动和外力项.若满足0(s)LPQds2.则有渐近稳定性当t,(t)-(t)J20.【英文摘要】One of the important contents in the dynamics is to study the infinite-dimensional dynamical systems of the atmosphere and the oceans.The understanding of the asymp-totic behavior of dynamical systems is one of the most important problems of modern mathematical physics.The thesis is devoted to considering the primitive equations of the atmosphere and the oceans.The model of the dissipative quasi-geostrophic equations was first given by Constantin:The thesis mainly considers the L2 decay,the high order decay and the asymptotic behavior.The thesis consists of three chapters.In the chapter 1, we introduce the background of the two dimensional dissipative quasi-geostrophic equation and preliminaries.In the chapter 2, using the spectral decomposition of the Laplacian operator and iterative techniques,we obtain improved L2 decay rates of weak solutions and derive more explicit upper bounds of the higher order derivatives of solutions. If 00)and(x,t)is a weak solution,thenm(t)L2C(1+t)-m+2/2, for large t.In the chapter 3,this chapter mainly prove the asymptotic behavior,i.e.,(t)-(t)L20.We choose the test function()=-U()st(-)U()()dto help to obtain the auxiliary liary decay estimate lim supt(1+A)-2-/41/2(t)L22=0. The perturbed equation of the two dimensional dissipative quasi-geostrophic equation is Here n(x)和f(x,t)are large initial and the external force perturbations respectively.satisfies the condition0=(s)Lpqds2.We have asymptotic stability【关键词】二维耗散准地转方程 L2衰减性 渐近稳定性【英文关键词】the two-dimensional dissipative quasi-geostrophic equation L2 decay asymptotic stability【目录】二维耗散准地转方程解的渐近性摘要3-4ABSTRACT4-5第一章 绪论7-151.1 研究背景7-101.2 主要结论10-111.3 预备知识11-15第二章 时间衰减性15-262.1 L