【摘要】：本文考慮Benjamin-Bona-Mahony方程解的長時間行為.首先,研究具有周期邊界條件的二維廣義Benjamin-Bona-Mahony方程,采用正交分解方法證明漸近吸引子的存在性,從而克服了近似慣性流形的精度問題.最后,給出了漸近吸引子的維數估計.其次,研究半離散的n維廣義Benjamin-Bona-Mahony方程,先對時間進行Crank-Nicolson格式化,進而證明這個半離散化的廣義方程在H1(Rn)中擁有一個離散的無窮維動力系統(tǒng),且該系統(tǒng)在H1(Rn)中存在全局吸引子Aτ,并證明了全局吸引子Aτ是正則的.最后,給出了全局吸引子Aτ的有限分形維數估計.全文共分為三個部分:·第一章,主要介紹了帶有周期邊界的二維廣義Benjamin-Bona-Mahony方程和半離散的n維廣義Benjamin-Bona-Mahony方程的背景,以及發(fā)展方程解的長時間行為的基本理論和方法.·第二章,證明了二維廣義Benjamin-Bona-Mahony方程漸近吸引子的存在性,并且給出了該漸近吸引子的維數估計.·第三章,證明了半離散的n維廣義Benjamin-Bona-Mahony方程全局吸引子的存在性,并且給出了該全局吸引子的有限分形維數估計.
[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.
【學位授予單位】：西南大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175

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