CKN theory of singularities of weak solutions of the Navier-Stokes equations

TL;DR

This paper provides a comprehensive overview of the CKN theory, which analyzes the formation and structure of singularities in weak solutions of the Navier-Stokes equations within a periodic domain.

Contribution

It offers a complete exposition of the CKN theory, highlighting recent improvements and geometrical insights into singularity behavior in fluid dynamics.

Findings

Detailed description of singularity structure in Navier-Stokes solutions

Improved estimates on singularity measures

Geometrical interpretation of singularity formation

Abstract

The lectures are devoted to a complete exposition of the theory of singularities of the Navier Stokes equations solution studied by Leray, in a simple geometrical setting in which the fluid is enclosed in a container $\O$ with periodic boundary conditions and side size $L$. The theory is due to the work of Scheffer, Caffarelli, Kohn, Nirenberg and is called here CKN-theory as it is inspired by the work of the last three authors which considerably improved the earlier estimates of Scheffer.