Well-posedness of rough 2D Euler equation with bounded vorticity

TL;DR

This paper proves the well-posedness and stability of the 2D Euler equation with bounded vorticity under rough transport noise, establishing uniqueness and a Wong-Zakai approximation for fractional Brownian paths.

Contribution

It demonstrates the existence, uniqueness, and stability of solutions to the 2D Euler equation with rough noise, extending classical results to stochastic and rough path settings.

Findings

Unique solution exists for the rough 2D Euler equation with bounded vorticity.

Solution map is stable with respect to initial vorticity and noise perturbations.

Wong-Zakai approximation holds for fractional Brownian driving paths.

Abstract

We consider the 2D Euler equation with bounded initial vorticity and perturbed by rough transport noise. We show that there exists a unique solution, which coincides with the starting condition advected by the Lagrangian flow. Moreover, the stability of the solution map with respect to the initial vorticity and the rough perturbation yields a Wong-Zakai result for fractional Brownian driving paths.