On the Lamb vector divergence, evolution of pressure fields and Navier-Stokes regularity

TL;DR

This paper investigates the divergence of the Lamb vector and its role in maintaining the regularity of the Navier-Stokes equations by defining a special pressure field through elliptic PDEs that acts as a control potential.

Contribution

It introduces a method to choose a pressure field that guarantees Navier-Stokes regularity, based solely on the velocity field and its derivatives, applicable to divergence-free initial data.

Findings

Pressure field can be chosen to ensure system regularity.

The pressure field is defined via elliptic PDEs.

The approach is applicable to general divergence-free initial conditions.

Abstract

This paper analyzes the Lamb vector divergence, also called the hydrodynamic charge density, and its implications to the Navier-Stokes system. It is shown that the pressure field can be always chosen in a way that ensures regularity of the Navier-Stokes system. The abstract pressure field that ensures regularity is defined through two partial differential equations, being of the elliptic kind. The pressure field defined such a way can be interpreted as a control potential field that keeps the system regular. The controlling pressure field depends only on the velocity field of the fluid and its derivatives, so that the result is applicable in any general setting where the initial data is divergence free, smooth and square-integrable.