【作者】 高英；

【导师】 田立新；

【作者基本信息】 江苏大学 ， 应用数学， 2009， 硕士

【摘要】 非线性现象是自然界中普遍存在的一种重要现象。最近几十年来,物理、力学、化学、生物、工程、航空航天、医学、经济和金融等领域中诞生了许多非线性偏微分方程,但是由于方程的非线性以及本身的复杂性,使得对这些方程的研究具有很大的挑战性。本文研究了两类有着深刻物理背景的非线性偏微分方程,即粘性Fomberg-Whitham方程和一类粘性色散波方程。本文研究的主要内容:引进整体吸引子的概念,考虑在周期边界条件下F0nlberg-Whitham方程和一类粘性色散波方程的全局解和全局吸引子存在性问题。第三章研究带粘性项的Fornberg-Whitham方程,运用伽辽金方法得到了L~2(R)空间下全局解的存在性,结果表明了在L~2(R)空间中F0rnberg-Whitham方程存在唯一的全局解。接着利用Sobolev插值不等式以及利用关于时间t的先验估计等方法证明了该方程在H~2(R)空间上吸收集的存在性,最后通过证明方程的解半群S(t)是一个紧算子得到Fornberg-Whitham方程全局吸引子的存在性。第四章研究了一类粘性色散波方程。这一章用了和第三章一样的讨论方法。首先得到了全局解的存在性,接着讨论了方程解半群吸收集在H~2(R)空间上的存在性,最后证明了该方程存在全局吸引子。更多还原

【Abstract】 Nonlinearity is universal and important phenomenon in nature. In recent years, many nonlinear partial differential equations were derived from physics, mechanics, chemistry, biology, engineering, aeronautics, medicine, economy, finance and many other fields. Because of the non-linearity and complexity of themselves, it is a big challenge to deal with them. In the paper, we study two nonlinear partial differential dispersive equations, that is, viscous Fornberg-Whitham equation and a class of viscous nonlinear dispersive wave equations. In this paper, we introduce the concept of global attractor and get the existence of global solution and the global attractor of a viscosity Fornberg-Whitham equation and the viscous nonlinear dispersive wave equation on periodical boundary condition.In the third chapter, the Galerkin Procedure is applied to show the existence of the global solution of Fornberg-Whitham equation in L~2(R). The Sobolev interpolation inequality and prior estimate on time-t are applied to show the existence of attracting set. Moreover, we prove the semi-group of the solution operator is a compact operator. Finally, we get the existence of the global attractor of a viscous Fornberg-Whitham equation in H~2(R).In the forth chapter, we give a study on a class of viscous nonlinear dispersive wave equations. We use the same discussion method as the third chapter. First we show the existence of global solution of the viscous nonlinear dispersive wave equation, then the existence of attracting set is obtained in H~2(R). Finally we proof that the equation has a global attractor.更多还原