导　　师： 郭柏灵

学科专业： 070104

授予学位： 博士

作　　者： ;

机构地区： 中国工程物理研究院

摘　　要： 本文考察了KdV、BBM、GBBM、KdV-Burgers、广义耦合的非线性波动方程组等非线性发展方程整体光滑解及其渐近行为，利用先验估计，对一类广义KdV方程组及耦合的波动方程组的周期初值问题、Cauchy问题、初边值问题进行了讨论，研究了整体吸引子的存在性及其分形维数有限性估计。 本文由六章组成： 第一章，给出了KdV、BBM、GBBM、KdV-Burgers、广义耦合的非线性波动方程组等非线性发展方程的的物理背景，回顾了已有的一些重要结果，简述了本文主要的研究结果。 第二章，考虑一类一维非齐次BBM方程，在第二节中利用Fourier谱方法和先验估计证明了具有周期初值问题的整体光滑解的存在性和唯一性，给出了Fourier谱近似解和精确解的长时间误差估计；在第三、四节中讨论了初边值问题，利用与时间t无关的一致先验估计，证明了整体光滑解和整体吸引子的存在性。 第三章，考虑高维的非齐次GBBM方程的初边值问题，建立与时间t无关的一致先验估计，证明了整体光滑解和整体吸引子的存在性。 第四章，考虑一类具耗散的广义KdV方程组的周期初值问题，在第二节中证明了整体光滑解的存在性和唯一性，得到整体吸引子；在第三节中构造了半离散和全离散的Fourier谱格式和拟谱格式，在整体光滑解存在的条件下，证明了这些格式解的收敛性，并得到了误差估计。 第五章，考虑了一类广义耦合的非线性波动方程组，在第二节中讨论了周期初值问题，证明了整体光滑解的存在性和唯一性，得到了整体吸引子，给出了Hausdorff维数和分形维数的上界估计；在第三节中讨论了Cauchy问题，利用加权函数和加权空间的插值不等式，证明了无界区域上整体吸引子的存在性；在第四节中证明了时间周期解的存在性。 第六章，考虑了一类广义耦合的KdV-Burgers方程，在第二节中讨论了半无界区域上的初边值问题，证明了整体光滑解和整体吸引子的存在性；在第三节中讨论了Cauchy问题，利用加权函数和加权空间上的插值 8 不等式，证明了半无界区域上整体吸引子的存在性。 本文的主要特点和难点在于作高维问题、非线性方程组问题及其无 界区域问题的先验估计时，都遇到了许多用常规方法难以克服的困难。 因此，针对不同问题，我们采用一系列复杂的、细致的先验估计，解决了 以上困难。 In this paper, we consider the global smooth solutions and long time be-haviors for some nonlinear evolution equations, such as KdV equation, BBM equation, GBBM equation, KdV-Burgers equation, coupled generalized nonlin-ear wave equation. By using a priori estimates, the existence of global smooth solution, the existence of global attractors and its fractal dimensions for this sys-tems are obtained. This paper is organized in six chapters. Chapter 1, give the physical background for the nonlinear evolution equa-tions, such as KdV equation, BBM equation, GBBM equation, KdV-Burgers equation, coupled generalized nonlinear wave equation. We look back to some important results, review our work for doing. Chapter 2, consider a class non-homogeneous BBM equation. In section 2.2, by a priori estimates and Fourier spectral method, we prove the existence and uniqueness of the global smooth solution for the periodic initial value problem and obtain the large time error estimate between spectral approximate solution and the exact solution. In sections 2.3 and 2.4, by a priori estimates and Galerkin method, we prove the existence of the global smooth solution and global attrac-tors for the initial-boundary value problem. Chapter 3, consider the initial-boundary value problem of the multidimen-sional non-homogeneous GBBM equations. The existence of global smooth so-lution and global attractors of this problem was proved by means of a uniform priori estimate for time. Chapter 4, consider the periodic initial value problem of a dissipative gen-eralized KdV equations. In section 4.2, the existence of global smooth solution of this problem is proved and the existence of the global attractors is obtained. In section 4.3, semi-discrete and fully discrete Fourier spectral and pseudo-spectral schemes are constructed. The convergence and stability for the schemes are proved, and the error estimates are obtained. Chapter 5, consider the damped coupled generalized nonlinear wave equations. In section 5.2, by coupled a priori estimates and Galerkin method, prove the existence and uniqueness of the global smooth solution for the periodic initial value problem and obtain the existence of global attractors. We get the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractors. In section 5.3, the Cauchy problem is studied, by using the weighted function space and the interpolating inequality, the existence of the global attractors for the damped generalized coupled nonlinear wave equations in an unbounded domain is proved. In section 5.4, the time periodic solution problem of damped generalized coupled nonlinear wave equations with periodic boundary conditions is studied, the existence of time periodic soluation of this problem is proved by using the convergence of approximate time periodic solution sequences. Chapter 6, consider a coupled generalized KdV-Burgers equation. In section 6.2, we study the initial-boundary value problem in the semi-unbounded domain, the existence of global solutions and global attractors is proved by means of a uniform priori estimate for time. In section 6.3, the Cauchy problem by using the weighted space, the existence of the global attractors for a coupled generalized KdV-Burgers in an semi-unbounded domain is proved. In this paper, the main difficulties are from a priori estimates for studing the high dimension, nonlinear systems and unboundary domain, we meet many problems which are difficult to be overcomed by using the standed method. We solve these difficulties by using the complicated, meticulous a priori estimates.

关 键 词： 非线性发展方程 耦合波动方程组 方程 方程 方程 方程 整体吸引子 谱和拟谱 先验估计 误差估计 无界区域

分 类 号： [O175.2]

领　　域： [理学] [理学]