Higher derivatives estimate for the 3D Navier-Stokes equation

TL;DR

This paper introduces a new family of function spaces based on energy dissipation to derive uniform estimates on higher derivatives of solutions to the 3D Navier-Stokes equations, valid up to potential blow-up times.

Contribution

It develops a novel framework connecting energy and critical spaces to obtain uniform higher derivative estimates for Navier-Stokes solutions.

Findings

Uniform estimates on higher derivatives up to blow-up time

Introduction of a new family of energy-based function spaces

Use of blow-up techniques and Galilean invariance

Abstract

In this article, a non linear family of spaces, based on the energy dissipation, is introduced. This family bridges an energy space (containing weak solutions to Navier-Stokes equation) to a critical space (invariant through the canonical scaling of the Navier-Stokes equation). This family is used to get uniform estimates on higher derivatives to solutions to the 3D Navier-Stokes equations. Those estimates are uniform, up to the possible blowing-up time. The proof uses blow-up techniques. Estimates can be obtained by this means thanks to the galilean invariance of the transport part of the equation.