一类非齐次拟线性双曲型方程组整体解的存在性
发布时间：2018-10-13 11:36
【摘要】：偏微分方程不论是在数学学科自身发展的需要方面,还是在其他自然学科中的应用方面,都是十分重要的。本文针对非齐次项既依赖于时间变量t又依赖于未知变量u的拟线性双曲型系统柯西问题的整体解存在性问题,在周忆教授的两个估计和正规化坐标以及匹配条件的基础上,得到了弱线性退化下波的分解公式,并得到其整体解的存在性。 第一章介绍了偏微分方程以及拟线性双曲型方程组在其他学科应用的重要性,介绍了研究拟线性双曲型方程组的重要意义及前人在这方面所做的贡献,提出本文的工作内容。 第二章主要介绍了在研究过程中常用的一些概念、定义。严格双曲型、弱线性退化、正规化坐标、匹配条件等,并给出了本文所讨论的主要问题。 第三章主要推导出了非齐次项既依赖于时间变量t,又依赖于未知变量u时的波的分解公式,该公式在后边定理的证明中起到了相当重要的作用。 第四章中主要引用了两个估计,为之后定理的证明和计算做了铺垫。 第五章通过波的分解公式和两个估计,利用一致先验估计的方法证明了满足弱线性退化条件时,非齐次拟线性双曲型方程组解的整体存在性。
[Abstract]:Partial differential equations (PDEs) are very important both in the needs of the development of mathematics and in the applications of other natural disciplines. In this paper, the existence of global solutions for the Cauchy problem of quasilinear hyperbolic systems with inhomogeneous terms dependent on both the time variable t and the unknown variable u is discussed. On the basis of Professor Zhou's two estimates and normalized coordinates and matching conditions, The decomposition formula of wave under weakly linear degeneracy is obtained, and the existence of its global solution is obtained. The first chapter introduces the importance of partial differential equations and quasilinear hyperbolic equations in other disciplines, introduces the significance of studying quasilinear hyperbolic equations and the contributions made by predecessors in this field, and puts forward the work contents of this paper. The second chapter mainly introduces some commonly used concepts and definitions in the research process. Strictly hyperbolic form, weakly linear degenerate, normalized coordinates, matching conditions, etc., and the main problems discussed in this paper are given. In the third chapter, we derive the decomposition formula of wave when the inhomogeneous term depends on both the time variable t and the unknown variable u, which plays a very important role in the proof of the theorem. In chapter 4, two estimators are cited, which pave the way for the proof and calculation of the following theorems. In chapter 5, the global existence of solutions of nonhomogeneous quasilinear hyperbolic equations is proved by means of the decomposition formula of wave and two estimators, using the method of uniform prior estimation.
【学位授予单位】：中北大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.27
本文编号：2268473
[Abstract]:Partial differential equations (PDEs) are very important both in the needs of the development of mathematics and in the applications of other natural disciplines. In this paper, the existence of global solutions for the Cauchy problem of quasilinear hyperbolic systems with inhomogeneous terms dependent on both the time variable t and the unknown variable u is discussed. On the basis of Professor Zhou's two estimates and normalized coordinates and matching conditions, The decomposition formula of wave under weakly linear degeneracy is obtained, and the existence of its global solution is obtained. The first chapter introduces the importance of partial differential equations and quasilinear hyperbolic equations in other disciplines, introduces the significance of studying quasilinear hyperbolic equations and the contributions made by predecessors in this field, and puts forward the work contents of this paper. The second chapter mainly introduces some commonly used concepts and definitions in the research process. Strictly hyperbolic form, weakly linear degenerate, normalized coordinates, matching conditions, etc., and the main problems discussed in this paper are given. In the third chapter, we derive the decomposition formula of wave when the inhomogeneous term depends on both the time variable t and the unknown variable u, which plays a very important role in the proof of the theorem. In chapter 4, two estimators are cited, which pave the way for the proof and calculation of the following theorems. In chapter 5, the global existence of solutions of nonhomogeneous quasilinear hyperbolic equations is proved by means of the decomposition formula of wave and two estimators, using the method of uniform prior estimation.
【学位授予单位】：中北大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.27
【参考文献】
相关期刊论文 前1条
1 孔德兴;一阶拟线性双曲型方程组的Cauchy问题[J];复旦学报(自然科学版);1994年06期
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