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

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.