Rough perturbations of the Navier-Stokes system and the vorticity formulation

Martina Hofmanova, James-Michael Leahy, Torstein Nilssen

Published 2019-02-25Version 1

We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance from the balance of momentum and conservation of circulation. We present a framework for a well-posedness analysis of the derived system. In particular, we define an intrinsic notion of solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. We prove that, in two space dimensions, well-posedness and pathwise energy estimates hold. Moreover, we derive rough path continuity of the equation, which, in particular, gives a Wong-Zakai result for the case of driving signals given by Brownian paths. In dimension three, the noise is not enstrophy conservative and we establish existence of local in time solutions.