【摘要】：偏微分方程不論是在數(shù)學(xué)學(xué)科自身發(fā)展的需要方面,還是在其他自然學(xué)科中的應(yīng)用方面,都是十分重要的。本文針對非齊次項(xiàng)既依賴于時間變量t又依賴于未知變量u的擬線性雙曲型系統(tǒng)柯西問題的整體解存在性問題,在周憶教授的兩個估計和正規(guī)化坐標(biāo)以及匹配條件的基礎(chǔ)上,得到了弱線性退化下波的分解公式,并得到其整體解的存在性。 第一章介紹了偏微分方程以及擬線性雙曲型方程組在其他學(xué)科應(yīng)用的重要性,介紹了研究擬線性雙曲型方程組的重要意義及前人在這方面所做的貢獻(xiàn),提出本文的工作內(nèi)容。 第二章主要介紹了在研究過程中常用的一些概念、定義。嚴(yán)格雙曲型、弱線性退化、正規(guī)化坐標(biāo)、匹配條件等,并給出了本文所討論的主要問題。 第三章主要推導(dǎo)出了非齊次項(xiàng)既依賴于時間變量t,又依賴于未知變量u時的波的分解公式,該公式在后邊定理的證明中起到了相當(dāng)重要的作用。 第四章中主要引用了兩個估計,為之后定理的證明和計算做了鋪墊。 第五章通過波的分解公式和兩個估計,利用一致先驗(yàn)估計的方法證明了滿足弱線性退化條件時,非齊次擬線性雙曲型方程組解的整體存在性。
[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.
【學(xué)位授予單位】：中北大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O175.27

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 孔德興;一階擬線性雙曲型方程組的Cauchy問題[J];復(fù)旦學(xué)報(自然科學(xué)版);1994年06期

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本文編號：2268473