博碩士論文 101221016 詳細資訊

本論文永久網址:   


以作者查詢圖書館館藏 以作者查詢臺灣博碩士 以作者查詢全國書目 勘誤回報 、線上人數:28 、訪客IP:3.144.96.70
姓名 林彥廷(Yen-ting Lin)  查詢紙本館藏   畢業系所 數學系
論文名稱
(Vector Fields With Given Vorticity, Divergence And The Normal Trace)
相關論文
★ Navier-Stokes 方程组弱解的存在性★ 用有限元素法解二維不可壓縮之拉格朗日式奈維-斯托克斯方程式的一些數值結果
★ 諾伊曼問題的二階線性雙曲方程之正則性理論★ A new approach of finding vector fields with prescribed divergence, vorticity and normal trace using the finite element method
檔案 [Endnote RIS 格式]    [Bibtex 格式]    [相關文章]   [文章引用]   [完整記錄]   [館藏目錄]   [檢視]  [下載]
  1. 本電子論文使用權限為同意立即開放。
  2. 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。
  3. 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) 對於一般的向量值函數$u$,我們有$u = curl w + abla p$的分解。我們證明了當函數$u$的旋度、散度與邊界法向量在三維球上給定並滿足可解條件時,$u$的存在性與唯一性。我們先考慮了在三維的全空間和上半空間對應問題之情況及求解方法,並從這些方法推得在三維的球上這個特殊情形下,另一種建構解的方式和一個與橢圓方程正則理論相似的正則性理論。
摘要(英) For a general vector-valued function $u$, we have the decomposition $u = curl w + abla p$. We proved the existence and uniqueness of $u$ when its vorticity, divergence and normal trace are prescribed in the unit ball of $bR^3$ under the assumption that the solvability condition holds. We start from solving for the velocity for the case that the domain under consideration is $bR^3$ or $bR^3_+$, and learn from this experience to provide another approach of constructing the solution and prove a regularity theory similar to the elliptic regularity theory.
關鍵字(中) ★ 向量場
★ 旋度
★ 散度
★ 邊界條件
關鍵字(英) ★ vector field
★ vorticity
★ divergence
★ boundary condition
論文目次 中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . i
英文摘要. . . . . . . . . . . . . . . . . . . . . . . . . ii
謝誌. . . . . . . . . . . . . . . . . . . . . . . . . . iii
目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1、Introduction . . . . . . . . . . . . . .. . . . . . . 1
1-1 The Equations . . . . . . . . . . . . . . . . . . . . 1
1-2 Previous Works . . . . . . . . . . . . .. . . . . . . 2
1-3 Reduction of the Problem . . . . . . . .. . . . . . . 3
1-4 The case Ω = R3 or R3_+. . . . . . . . . . . . . . . 3
1-4-1 The case Ω = R3 . . . . . . . . . . . . . . . . . . 4
1-4-2 The case Ω = R3_+ . . . . . . . . . . . . . . . . . 4
1-5 The Main Theorem . .. . . . . . . . . . . . . . . . . 7
1-6 Outlines . . . . . .. . . . . . . . . . . . . . . . . 7
2、Function Spaces and Mathematical Tools . . . . . . . . 8
2-1 The Sobolev Space Hs(Ω) and Some of Its Properties . 8
2-2 Lax-Milgram Theorem . . . . . . . . . . . . . . . . . 9
2-3 Poincaré-type inequality . . . . . . . . . . . . . . 10
2-4 Commutation with mollifiers . . . . . . . . . . . 11
2-5 The Piola Identity . . . . . . . . . . . . . . . . . 13
3、A Transfomation of the Origianl Problem . . . . . . . 14
3-1 Differential operators in spherical coordinate . . . 14
3-2 An Equivalent Problem of Equation (1.5) . . . .. . . 15
3-2-1 The boundary conditions . . . . . . .. . . . . . . 16
3-3 The Weak Formulation of Equation (3.8) . . . . . . 16
3-4 The Existence and Uniqueness of the Weak Solution to Equation(3.8) . . . . . . . . . . . . . . . . . . . . . 17
4、Existence, Uniqueness and Regularity of the Solution to (1.1) . . . . . . . . . . . . . . . . . . . . 18
4-1 The Regularity of w . . . . . . . . . . . . . . . . 18
4-1-1 Interior estimates . . . . . . . . . . . . . . . . 18
4-1-2 Boundary estimates of ∂ ℓ in normal direction . . 20
4-1-3 Boundary estimates of ∂ ℓ in tangential direction 25
4-1-4 Full gradient estimates . . . . . . . . . . . . . 26
4-2 The weak solution w to (3.8) has zero divergence . . 27
4-3 The Proof of the Main Theorem . . . . . .. . . . . . 28
References . . . . . . . . . . . . . . . . . . . . . . . 29
參考文獻 [1] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for
solutions of elliptic partial differential equations satisfying general boundary
conditions. II. Comm. Pure Appl. Math., 17:35–92, 1964.
[2] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in
three-dimensional non-smooth domains. Math. Methods Appl. Sci., 21(9):823–
864, 1998.
[3] Chérif Amrouche and Vivette Girault. Decomposition of vector spaces and
application to the Stokes problem in arbitrary dimension. Czechoslovak Math.
J., 44(119)(1):109–140, 1994.
[4] J. Aramaki. lp theory of the div-curl system. Int. J. Math. Anal., 8(6):259–271,
2014.
[5] Jürgen Bolik and Wolf von Wahl. Estimating ∇u in terms of div u, curl u,
either (ν, u) or ν × u and the topology. Math. Methods Appl. Sci., 20(9):737–
744, 1997.
[6] A. Buffa and P. Ciarlet, Jr. On traces for functional spaces related to Maxwell’s
equations. I. An integration by parts formula in Lipschitz polyhedra. Math.
Methods Appl. Sci., 24(1):9–30, 2001.
[7] A. Buffa and P. Ciarlet, Jr. On traces for functional spaces related to Maxwell’s
equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra
and applications. Math. Methods Appl. Sci., 24(1):31–48, 2001.
[8] Lawrence C. Evans. Partial differential equations, volume 19 of Graduate Studies
in Mathematics. American Mathematical Society, Providence, RI, second
edition, 2010.
[9] Hideo Kozono and Taku Yanagisawa. Lr-variational inequality for vector fields
and the Helmholtz-Weyl decomposition in bounded domains. Indiana Univ.
Math. J., 58(4):1853–1920, 2009.
[10] Mitchell A. R. Neittaanmäki P., Saranen J. Finite element approximation of
vector fields given by curl and divergence. Math. Methods Appl. Sci., 3(1):328–
335, 1981.
[11] Günter Schwarz. Hodge decomposition—a method for solving boundary value
problems, volume 1607 of Lecture Notes in Mathematics. Springer-Verlag,
Berlin, 1995.
[12] Wolf von Wahl. Estimating ∇u by div u and curl u. Math. Methods Appl. Sci.,
15(2):123–143, 1992.
指導教授 鄭經斅 審核日期 2014-8-29
推文 facebook   plurk   twitter   funp   google   live   udn   HD   myshare   reddit   netvibes   friend   youpush   delicious   baidu   
網路書籤 Google bookmarks   del.icio.us   hemidemi   myshare   

若有論文相關問題,請聯絡國立中央大學圖書館推廣服務組 TEL:(03)422-7151轉57407,或E-mail聯絡  - 隱私權政策聲明