A new proof of the Th\'eor\`eme de Structure related to a weak solution to the Navier-Stokes equations

TL;DR

This paper presents a new proof of the Th\'eor\`eme de Structure for Leray's weak solutions to the Navier-Stokes equations, extending the result to more general settings using a priori estimates.

Contribution

It introduces a novel proof method based on a priori estimates for approximating sequences, broadening the applicability to weak solutions without strong energy inequalities.

Findings

Established the Th\'eor\`eme de Structure with a new proof technique.

Extended the theorem to weak solutions in bounded domains without strong energy inequalities.

Utilized Galerkin approximation and a priori estimates for the proof.

Abstract

It is well known that a Leray's weak solution to the Navier-Stokes Cauchy problem enjoys a partial regularity which is known in the literature as the Th\'eor\`eme de Structure of a Leray's weak solution. As well, this result has been extended by some authors to the case of the IBVP. In this note, we achieve the Th\'eor\`eme de Structure by means of a new proof. Our proof is based on a priori estimates for a suitable approximating sequence. In this way our result covers a more general setting in the sense that, e.g., we can also include the case of the weak solutions furnished by Hopf for an IBVP in bounded domains without requiring an energy inequality in a strong form, but just employing a priori estimates on the Galerkin approximation.