An Elementary Solution of the Navier-Stokes Existence and Smoothness: Elementary Weak Solutions

This is an application of the functional analysis. There are no long or complicated calculations, semi-groups or apriori estimates or boundary conditions are not used at all. Our weak solutions do not satisfy energy inequality on <i>Ω</i>, but our solutions are smooth, unique if <i>u, v</i>∈<i>S</i> and <i>u-v</i>∈<i>S</i>, and the map <i>f→u </i>is continuous. The existence follows similar to the proof of Banach's fixed point theorem. The initial value <i>a</i>∈<i>C^</i>∞(Ω'), our solutions <i>u</i>∈<i>C</i>(<i>I</i>; <i>L^</i>2(Ω')).