Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

TL;DR

This paper establishes local-in-space regularity estimates near the initial time for weak solutions of the 3D Navier-Stokes equations, demonstrating the existence of smooth, scale-invariant solutions starting from homogeneous initial data.

Contribution

It introduces new local-in-space regularity estimates near initial time for weak solutions of Navier-Stokes, enabling the construction of global scale-invariant solutions from homogeneous initial data.

Findings

Existence of global scale-invariant solutions with homogeneous initial data

Local-in-space regularity estimates near initial time

Solutions are smooth for positive times

Abstract

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with $(-1)$-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are local-in-space regularity estimates near the initial time, which are of independent interest.