Navier-Stokes regularity in 3D

TL;DR

This paper presents a short proof demonstrating that smooth, fast-decaying initial data lead to eternal solutions of the 3D incompressible Navier-Stokes equations, confirming their regularity for all time.

Contribution

It introduces a novel proof technique using orthogonal decomposition and vector calculus identities to establish global regularity of Navier-Stokes solutions.

Findings

Solutions remain regular for all time with smooth, fast-decaying initial data

Enstrophy is shown to be non-increasing over time

The proof confirms the absence of finite-time singularities in this setting

Abstract

This short proof shows that for smooth and sufficiently fast decaying initial data at infinity, the full incompressible Navier-Stokes solutions are eternal. The proof uses an orthogonal decomposition of the velocity field and some well-known vector calculus identities to establish a particular contradiction, which leads to a vanishing integral, which is the main integral that determines the evolution of enstrophy. As it is shown that enstrophy is non-increasing, it is well-know that the solutions stay regular at all times.