Solution properties of the incompressible Euler system with rough path advection

The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider Euler for incompressible flow an ideal whose Lagrangian transport velocity possesses additional rough-in-time, divergence-free vector field. recent work, have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles possess a perturbative path Lie-advection constraint. paper, prove in L2-Sobolev spaces Hm with integer regularity m≥⌊d/2⌋+2 Beale-Kato-Majda (BKM) blow-up criterion terms Lt1Lx∞-norm vorticity. dimension two, show Lp-norms vorticity are conserved, which yields global Wong-Zakai approximation theorem stochastic version equation.