【作者】 金翠连；

【导师】 孔德兴；

【作者基本信息】 上海交通大学 ， 应用数学， 2008， 博士

【摘要】 本文的主要目的是研究拟线性双曲方程组整体解的存在唯一性.本文的主要内容由以下几章组成:第一章为绪言,在本章中我们对一阶拟线性双曲方程组Cauchy问题和Riemann问题的物理背景和研究现状做了一个简单介绍,并对本文要研究的几个问题加以阐述,叙述我们得到的结果.在第二章中,我们研究了具有弱线性退化特征的双曲平衡律方程组的Cauchy问题的整体经典解的存在性.证明了:如果初值适当的小,非齐次项或者满足强耗散条件,或者满足匹配条件,那么存在唯一的整体经典解.在第三章中,我们研究了一阶拟线性双曲守恒律方程组Riemann问题解的整体存在性.我们证明了,在第一象限内的混合初边值问题,特征要么是真正非线性的,要么是线性退化的,如果初值适当的小且满足一定的条件,则C1分片光滑解是整体唯一存在的,这个解的整体结构稳定,且相似于对应Riemann问题的解,当且仅当这个解仅包含激波和切触间断,而没有中心稀疏波或其它类型的弱间断.更多还原

【Abstract】 This thesis concerns with the global existence of smooth solu-tions of quasilinear hyperbolic systems of balance laws. The thesis isorganized as follows.Chapter 1 is an introduction. In this chapter, we simply recallthe physical background and present situation of the study on theCauchy problem and Riemann problem of quasilinear hyperbolic sys-tems of first order. We illustrate the problems which we shall discussand state the main results contained.In Chapter 2, we consider a kind of quasilinear hyperbolic sys-tems with inhomogeneous terms satisfying dissipative condition ormatching condition. For the Cauchy problem of this kind of systems,we prove that, if the system is weakly linearly degenerate and theinitial data is small and satisfies some decay condition, then theCauchy problem admits a unique global classical solution on t≥0.In Chapter 3, we investigate a class of mixed initial-boundaryvalue problems for a kind of n×n quasilinear hyperbolic systems ofconservation laws on the quarter plan. We show that the structure of the piecewise C~1 solution u = u(t,x) of the problem, which can beregarded as a perturbation of the corresponding Riemann problem, isglobally similar to that of the solution u = U(z/t) of the correspondingRiemann problem. The piecewise C~1 solution u = u(t,x) to this kindof problems is globally structure stable if and only if it contains onlynon-degenerate shocks and contact discontinuities, but no rarefactionwaves and other weak discontinuities.更多还原