Benjamin-Bona-Mahony方程解的长时间行为研究
发布时间：2018-07-14 18:30
【摘要】：本文考虑Benjamin-Bona-Mahony方程解的长时间行为.首先,研究具有周期边界条件的二维广义Benjamin-Bona-Mahony方程,采用正交分解方法证明渐近吸引子的存在性,从而克服了近似惯性流形的精度问题.最后,给出了渐近吸引子的维数估计.其次,研究半离散的n维广义Benjamin-Bona-Mahony方程,先对时间进行Crank-Nicolson格式化,进而证明这个半离散化的广义方程在H1(Rn)中拥有一个离散的无穷维动力系统,且该系统在H1(Rn)中存在全局吸引子Aτ,并证明了全局吸引子Aτ是正则的.最后,给出了全局吸引子Aτ的有限分形维数估计.全文共分为三个部分:·第一章,主要介绍了带有周期边界的二维广义Benjamin-Bona-Mahony方程和半离散的n维广义Benjamin-Bona-Mahony方程的背景,以及发展方程解的长时间行为的基本理论和方法.·第二章,证明了二维广义Benjamin-Bona-Mahony方程渐近吸引子的存在性,并且给出了该渐近吸引子的维数估计.·第三章,证明了半离散的n维广义Benjamin-Bona-Mahony方程全局吸引子的存在性,并且给出了该全局吸引子的有限分形维数估计.
[Abstract]:In this paper, we consider the long time behavior of the solution of Benjamin-Bona-Mahony equation. Firstly, the two-dimensional generalized Benjamin-Bona-Mahony equation with periodic boundary conditions is studied. The existence of asymptotic attractors is proved by orthogonal decomposition method, which overcomes the accuracy problem of approximate inertial manifolds. Finally, the dimension estimation of asymptotic attractor is given. Secondly, the semi-discrete n-dimensional generalized Benjamin-Bona-Mahony equation is studied. The time is formatted by Crank-Nicolson, and it is proved that the semi-discrete generalized equation has a discrete infinite dimensional dynamical system in H _ 1 (R _ n). Moreover, there exists a global attractor A 蟿 in H 1 (R n), and it is proved that the global attractor A 蟿 is regular. Finally, the finite fractal dimension estimation of the global attractor A 蟿 is given. The thesis is divided into three parts: chapter 1, the background of the two-dimensional generalized Benjamin-Bona-Mahony equation with periodic boundary and the semi-discrete n-dimensional generalized Benjamin-Bona-Mahony equation, and the basic theory and method of the long-time behavior of the solution of the evolution equation are introduced. The existence of asymptotic attractor for two-dimensional generalized Benjamin-Bona-Mahony equation is proved, and the dimension estimation of the asymptotic attractor is given. In chapter 3, the existence of global attractor for n-dimensional generalized Benjamin-Bona-Mahony equation is proved. The finite fractal dimension of the global attractor is estimated.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2122579
[Abstract]:In this paper, we consider the long time behavior of the solution of Benjamin-Bona-Mahony equation. Firstly, the two-dimensional generalized Benjamin-Bona-Mahony equation with periodic boundary conditions is studied. The existence of asymptotic attractors is proved by orthogonal decomposition method, which overcomes the accuracy problem of approximate inertial manifolds. Finally, the dimension estimation of asymptotic attractor is given. Secondly, the semi-discrete n-dimensional generalized Benjamin-Bona-Mahony equation is studied. The time is formatted by Crank-Nicolson, and it is proved that the semi-discrete generalized equation has a discrete infinite dimensional dynamical system in H _ 1 (R _ n). Moreover, there exists a global attractor A 蟿 in H 1 (R n), and it is proved that the global attractor A 蟿 is regular. Finally, the finite fractal dimension estimation of the global attractor A 蟿 is given. The thesis is divided into three parts: chapter 1, the background of the two-dimensional generalized Benjamin-Bona-Mahony equation with periodic boundary and the semi-discrete n-dimensional generalized Benjamin-Bona-Mahony equation, and the basic theory and method of the long-time behavior of the solution of the evolution equation are introduced. The existence of asymptotic attractor for two-dimensional generalized Benjamin-Bona-Mahony equation is proved, and the dimension estimation of the asymptotic attractor is given. In chapter 3, the existence of global attractor for n-dimensional generalized Benjamin-Bona-Mahony equation is proved. The finite fractal dimension of the global attractor is estimated.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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