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研究生： 許柏翰
Po-Han Hsu
論文名稱： 諾伊曼問題的二階線性雙曲方程之正則性理論
The regularity theory for second-order linear hyperbolic equation with Neumann boundary condition
指導教授： 鄭經斅
Ching-hsiao Cheng
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 100
語文別： 英文
論文頁數： 54
中文關鍵詞： 虛擬擴散 、正則性.
外文關鍵詞： artificial diffusion, regularity, Galerkin method.
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    • 這篇文章主要探討的是Neumann邊界問題之二階線性雙曲方程的正則性理論。首先要指出的一點是，Neumann邊界問題及Dirichlet邊界問題並無本質上的差異。亦即，定理的要求並不因為邊界條件之不同而改變。然而，縱使如此，Neumann問題在處理上仍舊比Dirichlet問題複雜地多，原因無他，即分部積分的邊界項多是不易控制的。
    我們使用三種不同的方式來建立雙曲方程的正則性理論。第一種即是大家最為熟悉的Galerkin方法。但我們發現這個方法的侷限性頗高，因此做了一些變形，而此番變形即為第二種方法。
    第三種方法則較為複雜些。大意是我們在雙曲方程之中加入虛擬擴散(artificial diffusion)項，將其轉為拋物方程，從而保證了逼近解的存在性及正則性。而我們所要確定的就是當虛擬擴散歸零時，方程解的正則性仍會保持。

    In this article, we consider the regularity theory for the second-order linear hyperbolic equation, especially with Neumann boundary condition. In fact, there is no intrinsic difference between Neumann boundary condition and Dirichlet boundary condition. That is to say, we cannot annihilate any assumption or reach a better conclusion even if we consider the equation with Nuemann boundary condition. However, the Neumann boundary is much complicated , that is, we will find that there are much more technical problems to be conquer.
    We start the article from the existence and uniqueness theorem and then we discuss the main part of this article, the regularity theory. We use three approaches. The first one is by the familiar Galerkin method. Secondly, we still use the Galerkin method but with some modification. Third, we add an artificial diffusion in our hyperbolic equation and then we may see it as a parabolic equation. Hence, we may apply the theory of parabolic equations. We study the equation with artificial diffusion instead of the original one. What we want is that if we improve the regularity of the equation with artificial diffusion, then the result will preserve as the artificial diffusion tends to zero.

    Contents 摘要 i Abstract ii 誌謝辭 iii Contents iv 1 Introduction 1 2 The Existence and Uniqueness 3 2.1 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Existence of weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Uniqueness of weak solution . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Regularity : I 13 3.1 The estimate of kutt(t)kL2( ) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Regularity : II 25 4.1 Fixed point argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Compatibility condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 L1H2 regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Higher regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 Regularity : III 37 5.1 The existence of u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 L1H2 regularity with smooth data . . . . . . . . . . . . . . . . . . . . . . 40 5.3 L1H2 regularity with data possessing limited regularity . . . . . . . . . . 45 5.4 L1H4 regularity with smooth data . . . . . . . . . . . . . . . . . . . . . . 45 5.5 L1H4 regularity with data possessing limited regularity . . . . . . . . . . 50 5.6 Higher regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Bibliography 54

    Lawrence C. Evans Partial Di erential Equations, American Mathematical Society 19 (1998).
    Michel Chipot Elliptic Equations: An Introductory Course, Birkhauser Advanced Texts (2000).

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