Solution properties of the incompressible Euler system with rough path advection

TL;DR

This paper establishes local well-posedness and blow-up criteria for incompressible Euler equations driven by rough paths, extending classical results to rough-in-time divergence-free vector fields and proving global results in 2D.

Contribution

It introduces a framework for analyzing Euler equations with rough path advection, proving local well-posedness and a BKM blow-up criterion, and demonstrates global well-posedness in 2D.

Findings

Proved local well-posedness in Sobolev spaces for rough path Euler equations.

Established a BKM blow-up criterion based on vorticity norms.

Showed conservation of vorticity norms in 2D leading to global well-posedness.

Abstract

The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work, we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in $L^2$-Sobolev spaces $H^m$ with integer regularity $m\ge \lfloor d/2\rfloor+2$ and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the $L^1_tL^\infty_x$-norm of the vorticity. In dimension two, we show that the $L^p$-norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.