Some new consequences of the CKN-theory

TL;DR

This paper explores new implications of the CKN-theory for Navier-Stokes solutions, demonstrating how singularities are confined and establishing conditions for global regularity and smoothness of solutions.

Contribution

It introduces novel consequences of the CKN-theory, including global regularity after finite time and simplified proofs of known regularity results.

Findings

Singularities occur only at cone tips where solutions are smooth.

Global regularity of Leray-Hopf solutions after finite time.

H1-regularity implies global existence and smoothness.

Abstract

It is a simple consequence of the Cafarelli-Kohn-Nirenberg theory that every possible singularity in a thin Haussdorff-measurable set of a Leray- Hopf solution of the incompressible Navier Stokes equation is on the tip of a small open cone, where the solution is smooth. Using global regularity results for weak Hopf-Leray solutions this potential singularity can be analyzed by investigation of the asymptotic behavior at infinite time of a solution of a related initial-boundary value problem posed in transformed coordinates on a cylinder. Next to some new consequences such as global regularity of the Leray Hopf solution after finite time, many known results can be recovered with this method succinctly, especially the result that H1-regularity implies global existence and smoothness.