可壓縮流中微黏性尤拉方程激波解的行為

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姓名

蘇承芳(Cheng-Fang Su) 查詢紙本館藏

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論文名稱

可壓縮流中微黏性尤拉方程激波解的行為
(Inner solutions for the viscous shock profiles of compressible Euler equations in a variable area duct)

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摘要(中)

本論文中，我們考慮的是在可變面積輸送管內的可壓縮、具微黏性之尤拉方程。藉著奇異擾動下的漸近展開式技術，我們可由黏性係數的階來研究此微黏性激波的內部解行為。此外，我們亦證明出O(1)與O(ε)之內部解方程可被修正成積分微分方程的形態，利用收縮映射原理，就可建立兩點邊界值問題解之存在性與唯一性。

摘要(英)

In this paper we consider the viscous compressible Euler equations in a variable area duct. By the technique of asymptotic expansions in singular perturbations, we study the inner solutions of the viscous shock profiles. The equations for inner solutions with respect to the power of viscous constant are derived. We show that the equations of inner solutions of O(1) and O(ε) can be modified to the scalar integro-differential equations. The existence and uniqueness of solutions for such two point boundary value problems are established by contraction mapping principle.

關鍵字(中)

★ 守恆律
★ 可壓縮尤拉方程
★ 微黏性激波
★ 奇異擾動
★ 內部解
★ 外部解

關鍵字(英)

★ inner solutions
★ conservation laws
★ viscous shock profiles
★ compressible Euler equations
★ outer solutions
★ singular perturbation

論文目次

中文摘要 --i
英文摘要 --ii
致謝 --iii
Contents --v
List of Figures --vi
Abstract --1
1 Introduction --2
2 Derivation of Equations for Inner Solutions --4
3 Equations of Traveling Waves and Integro-differential Systems --10
4 Existence and uniqueness of Solution to the Two Point Boundary Value Problem --15
4.1 Profiles of Traveling Waves --15
4.2 Existence and uniqueness of Solutions to the Two Point Boundary Value Problem of integro-differential Equations --18
References --26

參考文獻

[1] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2005.
[2] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), pp. 697-715.
[3] Jonathan Goodman, Zhouping Xin, Viscous Limits for Piecewise Smooth Solutions to System of Conservation Laws, Arch. Rational Mech. Anal. 121 (1992), pp. 235-265.
[4] J. M. Hong, An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by ``weaker than weak' solutions of the Riemann problem, J. Diff. Equ. 222 (2006), pp. 515-549.
[5] J. M. Hong, C. H. Hsu, Y. C. Su, Global solutions for initial-boundary value problem of quasilinear wave equations, J. Diff. Equ. 245 (2008), pp. 223-248.
[6] P. D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math. 10 (1957), pp. 537-566.
[7] J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York, Berlin (1983).
[8] Wolfgang Walter, Ordinary Differential Equations, Springer-Verlag, New York, Berlin (1998).
[9] B. Whitham, Linear and nonlinear waves. New York, John Wiley, 1974.
[10] S.-H. Yu, Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Rational Mech. Anal. 146 (1999), pp. 275-370.

指導教授

洪盟凱(John M. Hong)

審核日期

2009-5-23

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