【作者】 陈爱华；

【导师】 秦铁虎；

【作者基本信息】 复旦大学 ， 应用数学， 2008， 博士

【摘要】 粘弹性动力学的研究具有重要的理论意义,同时又具有很高的应用价值.20世纪50年代末到60年代初Colemann和Noll等系统地发展了具有记忆材料的本构理论,特别是Colemann 1964年发表的“具记忆材料的热动力学”[3]一文,对这一学科的发展起了重要影响.本文讨论本构方程为单积分形式且具有记忆特性的粘弹性材料,主要证明了三维非线性粘弹性动力学方程组的时间周期解和非平凡行波解的存在性.关于非线性粘弹性动力学方程组整体解的存在性已有许多结果,但是关于周期解和行波解的存在性的结果相对较少.1991年,Feireisl E.对粘弹性固体材料,利用粘性正则化和补偿紧性的方法,对一种特殊情形证明了一维非线性粘弹性动力学方程周期弱解的存在性.1992年,QinT.H.[26]通过对一般的线性积分-偏微分方程的研究,对粘弹性固体材料,证明了一维线性粘弹性动力学方程周期解的存在性.1997年,Qin T.H.[28]对粘弹性固体材料,证明了一维半线性粘弹性动力学方程周期解的存在性.同年,Qin T.H.[29]对粘弹性固体和流体材料,利用Galerkin方法,藉助于积分核奇性特点,证明了一维非线性粘弹性动力学方程周期弱解的存在性.这些结果所讨论的对象均是一维粘弹性方程,我们将其推广到一般的三维非线性粘弹性动力学方程组.1976年,Greenberg J.M.[9]对具有长时间记忆的粘弹性材料,利用单调迭代的方法,证明了一维非线性粘弹性动力学方程行波解的存在性.1988年,Liu T.P.[21]对具有衰减记忆的粘弹性材料,证明了一维非线性粘弹性动力学方程光滑行波解和非光滑行波解的存在性.2003年,Qin T.H.和Ni G.X.[31]对具有特殊积分核的粘弹性材料,利用他们提出的高阶迭代方法,证明了三维非线性粘弹性动力学方程组行波解的存在性,我们将这一结果推广到具有一般单积分形式本构关系的情况.本文的具体安排如下:首先在第一章,简单介绍了粘弹性动力学方程组有关问题的研究历史与现状以及本文的主要结果.在第二章,讨论一般三维非线性粘弹性动力学方程组具Dirichlet边界条件的周期解问题.给出粘弹性固体和粘弹性流体模型,利用Galerkin方法,并藉助于积分核奇性特点,得到了具有分数指数的Sobolev空间里的能量估计,进而利用Sobolev空间的插值定理和紧性定理,分别就固体和流体两种不同情况证明了该问题时间周期弱解的存在性.在第三章,对具有一般单积分形式本构关系的三维粘弹性动力学方程组,在一定假设下,当传播速度介于平衡弹性张量确定的波速和瞬时弹性张量确定的波速之间时,先利用高阶迭代方法证明行波解的局部存在性,然后利用Schauder不动点定理,证明行波解的整体存在性.更多还原

【Abstract】 The study on viscoelastic dynamic system is an important subject in both theory and applications.From the end of 1950s to the early of 1960s,Coleman and Noll et al developed the constitutive theory of viscoelastic materials with memory. Especially,the article "Thermodynamic of materials with memory"[3],given by Coleman in 1964,exerted a great influence on this domain.This thesis deals with viscoelastic materials,with memory,of which the constitutive equation is a single-integral law.The main aim of the thesis is to prove the existence of time-periodic solutions and nontrivial traveling wave solutions to the three-dimensional viscoelastic dynamic system.At present,there have been some important works on the existence of global solutions to the nonlinear viscoelastic dynamic system,but the works on the existence of periodic solutions and traveling wave solutions are lacking.In 1991,for viscoelastic solid materials,using viscosity regularization and compensated compactness method,Feireisl E.[8]proved the existence of periodic weak solutions to the one-dimensional nonlinear viscoelastic dynamic equation in a special case.In 1992,by studying a general linear integro-differential equation,Qin T.H. [26]proved the existence of periodic solutions to the one-dimensional linear viscoelastic dynamic equation in the case of viscoelastic solid materials.In 1997,Qin T.H.[28]proved the existence of periodic solutions to the one-dimensional semilinear viscoelastic dynamic system.Moreover,by Galerkin method,and employing the singularity of integral kernel,Qin T.H.[29]proved the existence of periodic weak solutions to the one-dimensional nonlinear viscoelastic dynamic system for viscoelastic solid and liquid materials,respectively.All these results dealt only with the onedimensional viscoelastic equation,we extend them to a general three-dimensional nonlinear viscoelastic system. In 1976,for the viscoelastic materials exhibiting long range memory,by monotonic methods,Greenberg J.M.[9]proved the existence of traveling wave solutions to the one-dimensional nonlinear viscoelastic dynamic equation.In 1988,for the viscoelastic materials with fading memory,Liu T.P.[21]proved the existence of smooth and nonsmooth traveling wave solutions to the one-dimensional nonlinear viscoelastic dynamic equation.In 2003,for the viscoelastic materials with a special integral kernel,by virtue of a higher-order iterative process,Qin T.H.and Ni G.X.[31]proved the existence of traveling wave solutions to the three-dimensional nonlinear viscoelastic dynamic system.We extend this result to the case where the constitutive equation is a general single-integral law.The arrangement of this thesis is as follows:First of all in Chapter 1,we give the development of viscoelasticity and main results in this paper.In Chapter 2,we discuss time periodic solutions to a general three-dimensional nonlinear viscoelastic system with Dirichlet boundary condition.By Galerkin method, and employing the singularity of integral kernel,we obtain the energy estimates in Sobolev spaces with fractional index.Moreover,by employing interpolation theorems and compactness theorems in Sobolev spaces,we show the existence of the solutions to the problem for viscoelastic solid and liquid materials,respectively.In Chapter 3,for the viscoelastic materials with the constitutive relation in a general single-integral law,under certain hypotheses,if the speed of propagation is between the speeds determined by equilibrium and instantaneous elastic tensors,in virtue of a higher-order iterative process and the Schauder’s fixed point theorem, we get the existence of nontrivial traveling wave solutions to the three-dimensional nonlinear viscoelastic dynamic system.更多还原