On the pathwise uniqueness of stochastic 2D Euler equations with Kraichnan noise and $L^p$-data

TL;DR

This paper extends the proof of pathwise uniqueness for stochastic 2D Euler equations with Kraichnan noise to include all initial data in $L^1 igcap L^p$ for any $p > 1$, removing previous restrictions.

Contribution

It generalizes the existing results by removing the restriction on the integrability exponent $p$, proving pathwise uniqueness for a broader class of initial data.

Findings

Pathwise uniqueness holds for all $L^1 igcap L^p$ initial data with $p>1$.

The result broadens the class of initial conditions for which uniqueness is guaranteed.

The proof removes previous constraints on $p$, enhancing the understanding of stochastic 2D Euler equations.

Abstract

In the recent work [arXiv:2308.03216], Coghi and Maurelli proved pathwise uniqueness of solutions to the vorticity form of stochastic 2D Euler equation, with Kraichnan transport noise and initial data in $L^1\cap L^p$ for $p>3/2$. The aim of this note is to remove the constraint on $p$, showing that pathwise uniqueness holds for all $L^1\cap L^p$ initial data with arbitrary $p>1$.