An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations
An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations
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DOI：
10.1142/s0219199799000183
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发表时间：
1999-03
影响因子：
1.6
通讯作者：
Jonathan C. Mattingly;Y. Sinai
中科院分区：
文献类型：
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作者：
Jonathan C. Mattingly;Y. Sinai
这里v是粘度，p是压力，f1，f2是外力的分量，外力可能与时间有关。由于我们的设置是周期性的，因此函数u1、u2、flp、f1和f2在x中都是周期性的。为了简单起见，我们把周期取为1。Leray（[Ler 34]）在全平面R上证明了（1）弱解的第一个存在唯一性定理。后来E. Hopf（参见[Hop 51]）。1962年，Ladyzenskaya证明了一般二维域强解的存在性和唯一性[Lad 69]。V. Yudovich，C.福亚斯河Teman，P. Constantin和其他人开发了强大的方法，为（1）描述的动力学提供了深刻的见解（见[Yud 89，Tem 79，Tem 95，CF 88]）。本文的目的是给出三个定理的初等证明。这些定理暗示了（1）光滑解的存在性和唯一性，并进一步阐明了动力学的耗散性质。我们还将讨论我们的技术在三维环境中可以提供什么。在二维中，考虑涡度ω（x1，x2，t）=<$u1（x1，x2，t）<$x2 − <$u2（x1，x2，t）<$x1是有用的。控制ω的方程具有以下形式（参见[CM 93，DG 95]）
Here ν is the viscosity, p is the pressure, and f1, f2 are the components of an external forcing which may be time-dependent. As our setting is periodic, the functions u1, u2, ∇p, f1, and f2 are all periodic in x. For simplicity, we take the period to be one. The first existence and uniqueness theorems for weak solutions of (1) were proven by Leray ([Ler34]) in whole plane R. Later these results were extended by E. Hopf (see [Hop51]). In 1962, Ladyzenskaya proved existence and uniqueness results for strong solutions for general two-dimensional domains [Lad69]. V. Yudovich, C. Foias, R. Teman, P. Constantin, and others developed strong methods which provided deep insights into the dynamics described by (1) (see [Yud89, Tem79, Tem95, CF88]). The purpose of this paper is to present elementary proofs of three theorems. These theorems imply the existence and uniqueness of smooth solutions of (1) and shed some additional light on the dissipative character of the dynamics. We will also discuss what our techniques can give in the three-dimensional setting. In two-dimensions, it is useful to consider the vorticity ω(x1, x2, t) = ∂u1(x1,x2,t) ∂x2 − ∂u2(x1,x2,t) ∂x1 . The equation governing ω has the form ( see [CM93, DG95] )