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研究生： 陳柏宇
Chen, Bo-Yu
論文名稱： Evolving a Knotted Kirchhoff Elastic Rod
Evolving a Knotted Kirchhoff Elastic Rod
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： 2019
畢業學年度： 107
語文別： 英文
論文頁數： 51
中文關鍵詞： Geometric flows 、Fourth-order problem 、Kirchhoff elastic rods 、Gâteaux derivative of Möbius knot energy
英文關鍵詞： Geometric flows, Fourth-order problem, Kirchhoff elastic rods, Gâteaux derivative of Möbius knot energy
DOI URL： http://doi.org/10.6345/NTNU201900361
論文種類： 學術論文
相關次數： 點閱：87  下載：8 
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    • 無中文摘要

    In this thesis, we study the problem on how to find an equilibrium configuration of a knotted Kirchhoff
    elastic rod by evolving it within the same knot class. The evolution is given by the $L^2$ gradient flow of
    the total energy composed of the Kirchhoff elastic energy and the Möbius knot energy. We would prove
    the long-time existence of smooth solutions for the gradient flow and discuss the asymptotic limit.

    0 Introduction . . . . . . . . . . . . . . . . . . . . . . 1 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . .2 1.1 Basic Notations for Curves and Frames . . . . . . . . .2 1.2 The Total Energy . . . . . . . . . . . . . . . . . . . 2 1.3 The $L^2$ Gradient Flow of the Total Energy . . . . . .4 1.4 Interpolation Inequalities for $\nabla^m_s\kappa$  and Their Polynomials . . . . . . . . . . . . . . . . .6 1.5 Interpolation Inequalities for $ artial^m_s\kappa$ . .8 1.6 Interpolation inequality of $\nabla^m_s(\tau\times\kappa)$ . . . . . . . . . . . . 11 2 Some Properties of the Gradient of the Möbius Energy . . 13 2.1 A Decomposition on $\mathcal{H_f}$ . . . . . . . . . . 13 2.2 The Estimate of the Main Term . . . . . . . . . . . . .14 2.3 The Estimate of the Remainders . . . . . . . . . . . . 18 2.3.1 The Estimate of $\nabla^m_s R_1$ . . . . . . . . . . 18 2.3.2 The Estimate of $\nabla^m_s R_2$ . . . . . . . . . . 28 2.4 The Estimate outside the Singularities . . . . . . . . 34 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . 41 3 Long-time Existence of the Flow . . . . . . . . . . . . .41 3.1 Estimate for $\frac{\text{d}}{\text{d}t}\left(\frac{1}{2}\int_I\! |\nabla^m_s \kappa|^2\ \text{d}s\right)+\int_I\!|\nabla^{m+2}_s\kappa|^2\ \text{d}s$. . . . . . . . . . . . . . . . . . . . . . . . .42 3.2 Main Theorem . . . . . . . . . . . . . . . . . . . . . 48 References. . . . . . . . . . . . . . . . . . . . . . . . 51

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