### 博碩士論文 952201025 詳細資訊

 姓名 姚文銘(Man-meng Io)  查詢紙本館藏 畢業系所 數學系 論文名稱 共振守恆律的擾動黎曼問題的古典解(Classical Solutions to the Perturbed Riemann Problem of Scalar Resonant Balance Law) 檔案 [Endnote RIS 格式]    [Bibtex 格式]    [檢視]  [下載]本電子論文使用權限為同意立即開放。已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。 摘要(中) 在這篇論文中，我們探討單一非線性平衡律的擾動黎曼問題的古典解。此平衡律等價於一個二乘二非線性平衡系統，而且是一個共振的系統。 透過特徵線的方法，我們建立擾動黎曼問題的古典解。經由此古典解的點態極限，我們並獲得對應之黎曼問題的解的自相似性。 摘要(英) In this paper we study the classical solutions to the perturbed Riemann problem of some scalar nonlinear balance law in resonant case. The equation with source term is equivalent to a 2×2 nonlinear balance laws as described in [6, 7], and it is a resonant system due to the fact that the speeds of waves in the solution to this 2×2 system coincide. The characteristic method in [8] is applied to construct the classical solutions of perturbed Riemann problem. Moreover, we show that, the pointwise limit of classical solutions, which are deﬁned as the measurable solutions to the corresponding Riemann problem (with singular source) of perturbed Riemann problem, are self-similar as described in [12]. 關鍵字(中) ★ 擾動黎曼問題★ 黎曼問題★ 非線性平衡律★ 特徵線法★ Lax's 方法★ 守恆律 關鍵字(英) ★ Perturbed Riemann problems★ Riemann problems★ Nonlinear balance laws★ Conservation laws★ Lax's method★ Characteristic method 論文目次 中文摘要 ………………………………………………………………i 英文摘要 ………………………………………………………………ii Acknowledgement ………………………………………………………iii 目錄 ……………………………………………………………………iv 圖目錄 …………………………………………………………………v 表目錄 ……………………………………………………………………vi 1. Introduction …………………………………………………………2 2. Classical solutions of perturbed Riemann problem …………5 3. Stability of perturbed Riemann solutions …………………17 References ………………………………………………………………23 參考文獻 [1] C. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Ind. Univ. Math. J. 26 (1977), 1097-1119. [2] C. Dafermos, Solutions of the Riemann problem for a class of conservation laws by the viscosity method, Arch. Ration. Mech. Anal., 52 (1973), 1-9. [3] G. Dal Maso, P. LeFloch and F. Murat, Deﬁnition and weak stability of nonconservative products, J. Math. Pure. Appl., 74(1995), 483-548. [4] Ronald J. DiPerna, Measure-Valued Solutions to Conservation Laws, Arch. Ration. Mech. Anal., (1985), 223-270. [5] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18(1956), 697-715. [6] J. M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by ”weaker than weaker” solutions of the Riemann problem, J. Diﬀ. Equations, 222(2006), 515-549. [7] J. M. Hong and B. Temple, A Bound on the Total Variation of the Conserved Quantities for Solutions of a General Resonant Nonlinear Balance Law, SIAM J. Appl. Math. 64, No 3, (2004), pp 625-640. [8] J. M. Hong, Y. Chang and S.-W. Chou, Generalized Solutions to the Riemann Problem of Scalar Balance Law with Singular Source Term, preprint. [9] E. Isaacson and B. Temple, Convergence of 2 × 2 by Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math. 55 (1995), pp 625-640. [10] K. T. Joseph and P. G. LeFloch, Singular limits for the Riemann problem: general diﬀusion, relaxation, and boundary condition, in ” new analytical approach to multidimensional balance laws”, O. Rozanova ed., Nova Press, 2004. 23[11] S. Kruzkov, First order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970), 217-273. [12] P. D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math., 10(1957), 537-566. [13] P. G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Comm. Partial Diﬀerential Equations, 13(1988), 669-727. [14] T. P. Liu, The Riemann problem for general systems of conservation laws, J. Diﬀ. Equations, 18(1975), 218-234. [15] T. P. Liu, Quaslinear hyperbolic systems, Comm. Math. Phys., 68(1979), 141-172. [16] C. Mascia and C. Sinestrari, The perturbed Riemann problem for a balance law, Advances in Diﬀerential Equations, 1996-041. [17] O. A. Oleinik, Discontinuous solutions of nonlinear diﬀerential equations, Amer. Math. Soc. Transl. Ser. 2, 26 (1957), 95-172. [18] C. Sinestrari, The Riemann problem for an inhomogeneous conservation law without convexity, Siam J. Math. Anal., Vo28, No1, (1997), 109-135. [19] C. Sinestrari, Asymptotic proﬁle of solutions of conservation laws with source, J. Diﬀ. and Integral Equations, Vo9, No3,(1996), 499-525. [20] M. Slemrod and A. Tzavaras, A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, Ind. Univ. Math. J. 38 (1989), 1047-1073. [21] J. Smoller, Shock waves and reaction-dﬀusion equations, Springer, New York, 1983. [22] A. Tzavaras, Waves interactions and variation estimates for self-similar zero viscosity limits in systems of conservation laws, Arch. Ration. Mech. Anal., 135 (1996), 1-60. [23] A. Volpert, The space BV and quasilinear equations, Maths. USSR Sbornik 2 (1967), 225-267. 指導教授 洪盟凱(John M. Hong) 審核日期 2008-7-21 推文 facebook   plurk   twitter   funp   google   live   udn   HD   myshare   reddit   netvibes   friend   youpush   delicious   baidu