簡易檢索 / 詳目顯示

回結果列表
研究生： 張育瑋
Chang, Yu-Wei
論文名稱： Smoothness of Local Solutions for the 4th-Order Elastic Flow of Inextensible Plane Curves
Smoothness of Local Solutions for the 4th-Order Elastic Flow of Inextensible Plane Curves
指導教授： 林俊吉
Lin, Chun-Chi
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2016
畢業學年度： 104
語文別： 英文
論文頁數： 56
中文關鍵詞： 非延伸性 、彈性流 、彎曲能 、平面曲線
英文關鍵詞： Inextensibility, Elastic flow, Bending energy, Plane curve
DOI URL： https://doi.org/10.6345/NTNU202204555
論文種類： 學術論文
相關次數： 點閱：132  下載：24 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 無中文摘要

    In this paper we establish the smooth short time existence of the inextensible elastic flow, provided that our initial plane curves is $C^{3+\alpha}$. We consider two cases of initial plane curves: closed curves and open curves with hinged end.
    The inextensible elastic flow of such plane curves correspond to semi-linear fourth-order parabolic equations.

    1 Introduction 2 2 Preliminaries and Notations 3 2.1 Holder Spaces 3 2.2 Parabolic Holder Spaces 4 3 Linear Theory 7 3.1 The Cauchy Problem 8 3.1.1 Construction of Biharmonic Heat Kernel 9 3.1.2 Properties of potential b \ast h and convolution b \ast_x u0 10 3.1.3 Proof of Theorem 3.1 and Theorem 3.2 21 3.2 The Initial-Boundary Value Problem for Hinged End 22 3.2.1 Construction of Poisson Kernels 23 3.2.2 Estimates of convolution on time 28 3.2.3 Proof of Theorem 3.9 and Theorem 3.10 35 4 The Inextensible Elastic Flow 38 4.1 Closed Curves 38 4.1.1 Estimates of ODE 38 4.1.2 Proof of Main Theorem 1.1 39 4.2 Open Curves for Hinged End 44 4.2.1 Estimates of ODE 45 4.2.2 Proof of Main Theorem 1.2 48 Appendix A Some Useful Lemmas 53 Appendix B Proof of Properties of Kernels 54 Appendix C Laplace transforms 55 Reference 56

    A. Ferrero, F. Gazzola, H.-Ch. Grunau. Decay and eventual local positivity for biharmonic parabolic equations. In Disc. Cont. Dynam. Syst. 21, 2008, 1129-1157.

    V.A. Galaktionov, S.I. Pohozaev. Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators. In Indiana Univ. Math. J. 51, 2002, 1321-1338.

    N. V. Krylov. Lectures on Elliptic and Parabolic Equations in Holder spaces. In Amer. Math. Soci., Graduate Studies in Mathematics Volume 12, 1996.

    N. Koiso. On the motion of a curve towards elastica. In Actes da la Table Ronde de Geometrie Di erentielle (Luminy 1992), Semin Congr. 1 Soc. Math. France, Paris, 1996, 403-436.

    V. Solonnikov. On the boundary value problems for linear parabolic systems of di erential equations of general form. In Proc. Steklov Inst. Math, volume 83, pages 1-162, 1965.

    J. R. Cannon. The One-Dimensional Heat Equation. Cambridge University Press, 1984.

    G. B. Folland. Real Analysis: Modern Techniques and their Applications, 2nd ed. New York: Wiley, 1999.

    L. Mejlbro. The Laplace Transformation I General Theory Complex Functions Theory a-4, 1st ed. Bookboon, 1999.

    下載圖示
    QR CODE