An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations

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. These results are essentially the same as those of [FT89], however our point of view and proofs are different. 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] )