On 3D Navier-Stokes equations: regularization and uniqueness by delays

TL;DR

This paper introduces a delay-based regularization of the 3D Navier-Stokes equations, proving existence, uniqueness, and regularity of solutions, and analyzing the limit as delay approaches zero.

Contribution

It presents a novel delay regularization method that ensures solution regularity and uniqueness for the 3D Navier-Stokes equations.

Findings

Solutions become regular with delay regularization

Existence and uniqueness of global weak solutions are established

The limit as delay approaches zero yields a weak solution of the original equations

Abstract

A modified version of the three dimensional Navier-Stokes equations is considered with periodic boundary conditions. A bounded constant delay is introduced into the convective term, that produces a regularizing effect on the solution. In fact, by assuming appropriate regularity on the initial data, the solutions of the delayed equations are proved to be regular and, as a consequence, existence and also uniqueness of a global weak solution is obtained. Moreover, the associated flow is constructed and the continuity of the semigroup is proved. Finally, we investigate the passage to the limit on the delay, obtaining that the limit is a weak solution of the Navier-Stokes equations.