On a Variational Approach to the Navier-stokes Equations

 (i) ∂v ∂t + divx (v ⊗ v) +∇xp = ν∆xv + f ∀(x, t) ∈ Ω× (0, T ) , (ii) divx v = 0 ∀(x, t) ∈ Ω× (0, T ) , (iii) v = 0 ∀(x, t) ∈ ∂Ω × (0, T ) , (iv) v(x, 0) = v0(x) ∀x ∈ Ω . Here v = v(x, t) : Ω× (0, T ) → R is an unknown velocity, p = p(x, t) : Ω× (0, T ) → R is an unknown pressure, associated with v, ν > 0 is a given constant viscosity, f : Ω× (0, T ) → R is a given force field and v0 : Ω → R N is a given initial velocity. The existence of weak solution to (1.1) satisfying the Energy Inequality was first proved in the celebrating works of Leray (1934). There are many different procedures for constructing weak solutions (see Leray [3],[4] (1934); Kiselev and Ladyzhenskaya [2] (1957); Shinbrot [5] (1973)). These methods are all based on the so called ”FaedoGalerkin” aproximation process. In this paper we give a new variational method to investigate the Navier-Stokes Equations. As an application of this method we give a new relatively simple proof of the existence of weak solutions to the problem (1.1). Let us briefly describe our method. Consider for simplicity f = 0 in (1.1). For every smooth u : Ω̄× [0, T ] → R satisfying conditions (ii)− (iv) of (1.1) define the energy functional (1.2) E(u) := 1 2 ∫ T