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研究生： 林彥廷
Yen-ting Lin
論文名稱： Vector Fields With Given Vorticity, Divergence And The Normal Trace
指導教授： 鄭經斅
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2014
畢業學年度： 102
語文別： 英文
論文頁數： 42
中文關鍵詞： 向量場 、旋度 、散度 、邊界條件
外文關鍵詞： vector field, vorticity, divergence, boundary condition
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    • 對於一般的向量值函數$\u$，我們有$\u = \curl \w + \nabla p$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。

    For a general vector-valued function $\u$, we have the decomposition $\u = \curl \w + \nabla p$. We proved the existence and uniqueness of $\u$ when its vorticity, divergence and normal trace are prescribed in the unit ball of $\bbR^3$ under the assumption that the solvability condition holds. We start from solving for the velocity for the case that the domain under consideration is $\bbR^3$ or $\bbR^3_+$, and learn from this experience to provide another approach of constructing the solution and prove a regularity theory similar to the elliptic regularity theory.

    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

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