本論文是針對 Giga 及 Kambe 在渦度方程上研究結果的整理。在本論文中，我們將從描述帶有黏性且不可壓縮的流體運動的 Navier-Stokes 方程出發導出渦度方程。建基於渦度方程解的存在與唯一性，我們證明其解的漸進定理，即當時間足夠大時解的漸進行為。然後討論此漸進定理的一個應用。

This thesis is a survey of research result of Giga and Kambe on the vorticity equation. In this thesis, we start with the Navier-Stokes equations, which describe the incompressible viscous flows in fluid mechanics, then the vorticity equation is derived from them. Based on the existence and uniqueness for the solution of the vorticity equation, we have the asymptotic formula for the solution which means the solution will approach the fundamental solution of the heat equation as time is large enough. Then we discuss an application of this asymptotic formula.

Contents
1 Introduction 1
2 The Asymptotic Formula for VE 3
2.1 The Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . 3
2.2 The Two-dimensional Vorticity Equation . . . . . . . . . . . . . . . 3
2.3 Unique Existence Theorem for VE . . . . . . . . . . . . . . . . . . 8
2.4 Estimates for Solutions of VE . . . . . . . . . . . . . . . . . . . . . 13
2.5 Proof of the Asymptotic Formula . . . . . . . . . . . . . . . . . . . 30
3 Application: The Burgers Vortex 39
4 Inequalities and Theory for Heat Equations 41
Reference 44

References
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Euler equations”. Arch. Rational Mech. Anal., 128, 329-358
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Differential Equations: Asymptotic Behavior of Solutions and Self-Similar
Solutions”. Springer
[3] Y. Giga, T. Miyakawa, and H. Osada (1988), “Two dimensional Navier-Stokes
flow with measures as initial vorticity”. Arch. Rational Mech. Anal., 104, 223-
250
[4] T. Kambe (1984), “Axisymmetric vortex solution of Navier-Stokes equation”.
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R2 with a measure as the initial vorticity”. Diff. Integral Equations, 7, 949-966
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