This is an application of functional analysis to the existence and smoothness of the Navier–Stokes equations using elementary weak solutions in Sobolev spaces. We solve the problem in mathematics. The problems are not in physics, so we do not use any physics or assumptions-falsified mathematics. We use mathematics only. We can solve the problem by using an exactly and completely FALSIFIED resolution, where large initial values destroy the earth, because uniqueness does NOT hold, or SMALL initial values love your cup of coffee. There are no long or complicated culculations, or no theory of evolution equations. A Solution of a non-linear differential equation F(u)＝0 is a fixed point u＝G(u). The resolutions such as other papers which can't interpret in words of fixed point theorems are falsified. We solve ∂ₜ u －△u ＝Pf －P(uₙ・▽)uₙ, ∂ₜ u －△u ＝Pf －P(u・▽)u in X＝∩_(m≥5, p＝1, 2)W^(m, p)(Ω) (as a set). We construct u＝Φ[u]∈S, where S is a closed subset of X. By new fixed point theorem, existence of solutions holds even if Φ[S]⊂S does not hold. Our solutions are weak solution: 〈∂ₜ u －△u, φ〉 ＝〈f －▽p－(u・▽)u, φ〉 for any φ∈D_σ(Ω), because〈PU, φ〉＝〈U, φ〉, 〈div(u), φ〉＝0 for any φ∈D(Ω). Uniqueness follows without boundary conditions. Smoothness, the energy inequality also follow. The motions of fluids occur by an external force and motions by fluids itselves, so our solutions are physically suitable. Our Elementary PDE Theory also can be applied to non-linear heat equation, non-linear Laplace equation, non-linear wave equation, non-linear Schrödinger equation, compressible Navier-Stokes equations and complex Monge-Ampére equation. Japanese people use X to study hi-level mathematics, and this manuscript has been read by Naofumi Mori (@n_mori00), Associate Professor of Mathematical Sciences. This paper will be our thesis of Open University of Japan.