$\rho$-white noise solution to 2D stochastic Euler equations

TL;DR

This paper introduces a stochastic 2D Euler equation model with transport noise, using an approximation with random vortices, and establishes properties of the solution's density relative to the enstrophy measure.

Contribution

It develops a novel stochastic framework for 2D Euler equations with transport noise and proves a gradient estimate for the solution density, advancing understanding of stochastic fluid dynamics.

Findings

Constructed stochastic solutions via vortex approximation

Proved the density satisfies a weak continuity equation

Established a gradient estimate for the solution density

Abstract

A stochastic version of 2D Euler equations with transport type noise in the vorticity is considered, in the framework of Albeverio--Cruzeiro theory [1] where the equation is considered with random initial conditions related to the so called enstrophy measure. The equation is studied by an approximation scheme based on random point vortices. Stochastic processes solving the Euler equations are constructed and their density with respect to the enstrophy measure is proved to satisfy a continuity equation in weak form. Relevant in comparison with the case without noise is the fact that here we prove a gradient type estimate for the density. Although we cannot prove uniqueness for the continuity equation, we discuss how the gradient type estimate may be related to this open problem.