Vector Fields With Given Vorticity, Divergence And The Normal Trace

Vector Fields With Given Vorticity, Divergence And The Normal Trace

林彥廷(Yen-ting Lin)

指導教授 ： 鄭經斅

國立中央大學/理學院/數學系/碩士(2014年)

摘要

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。

關鍵字

向量場 ； 旋度 ； 散度 ； 邊界條件

並列摘要

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.

並列關鍵字

vector field ； vorticity ； divergence ； boundary condition

參考文獻

solutions of elliptic partial differential equations satisfying general boundary

conditions. II. Comm. Pure Appl. Math., 17:35–92, 1964.

[4] J. Aramaki. lp theory of the div-curl system. Int. J. Math. Anal., 8(6):259–271,

equations. I. An integration by parts formula in Lipschitz polyhedra. Math.

Methods Appl. Sci., 24(1):9–30, 2001.