On the dual systems to Euler and Navier-Stokes equations in R3

Abstract

In this paper we consider two model systems of equations obtained by taking the Hodge duals of the Euler and the Navier–Stokes systems, respectively. The ‘vorticity’ formulation of the new system becomes a nonlinear, non-local scalar equation having similarities to the quasi-geostrophic equations. We prove the local existence of classical solutions, and derive the Beale-Kato-Majda type of blow-up criterion for the dual Euler system. Moreover, we actually show that the finite-time blow-up occurs for generic non-trivial initial data for this system. For the dual Navier-Stokes equations we construct an explicitly global-in-time smooth solution for general initial data, using the Hopf-Cole transform. In the appendix we formulate the Euler system and the dual Euler system in terms of the Clifford algebra.