Duality, Vector advection and the Navier-Stokes equations

TL;DR

This paper explores the self-duality of the 3D vector advection equation, deriving classical Navier-Stokes results, and introduces a non-unique Feynman-Kac type formula indicating multiple flow solutions conserving vorticity.

Contribution

It establishes the self-duality of the 3D vector advection equation and derives a non-unique Feynman-Kac formula, connecting to classical Navier-Stokes existence results.

Findings

Proves the self-duality of the 3D vector advection equation.

Derives a Feynman-Kac type formula for the vector advection equation.

Shows existence of flows with conserved vorticity differing from standard flows.

Abstract

In this article we show that three dimensional vector advection equation is self dual in certain sense defined below. As a consequence, we infer classical result of Serrin of existence of strong solution of Navier-Stokes equation. Also we deduce Feynman-Kac type formula for solution of the vector advection equation and show that the formula is not unique i.e. there exist flows which differ from standard flow along which vorticity is conserved.