When Pythagoras meets Navier-Stokes
Igor Honoré
TL;DR
The paper addresses obtaining energy and vorticity estimates for the incompressible Navier–Stokes equations in $\mathbb{R}^d$ via a novel time‑decomposition method. It introduces a freezing‑flow based proxy and a Duhamel representation, then slices the time interval to exploit short‑time regularization and derive a strict $L^2$ energy bound $\|u(T,\cdot)\|_{L^2} \le \|u_0\|_{L^2} + \int_0^T \|f(s,\cdot)\|_{L^2} ds$, while explaining why the same argument cannot be extended to $L^p$ with $p\neq 2$ due to growth in the iteration. In 3D it further establishes a vorticity bound $\|\nabla\times u(T,\cdot)\|_{L^1}$ by adapting the approach to the vorticity equation and applying a similar time decomposition to control the nonlinear terms. The results illuminate the central role of the $L^2$ energy structure in ensuring stability and provide a framework for analyzing NS dynamics through a localized, kernel‑based representation. These insights contribute to a rigorous approach to energy dissipation and vorticity propagation in incompressible flows with potential implications for regularity studies and numerical analyses.
Abstract
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 $L^\infty ([0,T],L^2(\mathbb{R}^d,\mathbb{R}^d))$ 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 $L^p(\mathbb{R}^d,\mathbb{R}^d)$, $p \neq 2$. We also deduce, by similar arguments, an already known $L^\infty ([0,T],L^1(\mathbb{R}^3,\mathbb{R}^3))$ control of vorticity for $d=3$.