When Pythagoras meets Navier-Stokes

In this article, we develop a new method, based on a time decomposition of a Cauchy problem elaborated in [6], to retrieve the well-known $L^{\mathrm{\infty }}([0,T],L^2({\mathbb{R}}^d,{\mathbb{R}}^d))$ control of the solution of the incompressible Navier-Stokes equation in ${\mathbb{R}}^d$. We precisely explain how the Pythagorean theorem in $L^2({\mathbb{R}}^d,{\mathbb{R}}^d)$ allows to get the proper energy estimate; however such an argument does not work anymore in $L^p({\mathbb{R}}^d,{\mathbb{R}}^d)$, $p\mathrm{\ne }2$. We also deduce, by similar arguments, an already known $L^{\mathrm{\infty }}([0,T],L^1({\mathbb{R}}^3,{\mathbb{R}}^3))$ control of vorticity for $d=3$.