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# Dynamics of Partial Differential Equations

## Volume 6 (2009)

### Number 1

### Duality, vector advection and the Navier-Stokes equations

Pages: 53 – 93

DOI: https://dx.doi.org/10.4310/DPDE.2009.v6.n1.a4

#### Authors

#### 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.

#### Keywords

Navier-Stokes equations, Feynman Kac formula, vector advection

#### 2010 Mathematics Subject Classification

35Q30, 60H30, 76D05

Published 1 January 2009