Weak vorticity formulation of 2D Euler equations with white noise initial condition

TL;DR

This paper develops a weak vorticity formulation for 2D Euler equations with Gaussian initial conditions, constructing solutions as limits of random point vortices and extending results to broader initial measures.

Contribution

It introduces a new weak vorticity approach for 2D Euler equations with stochastic initial data, connecting point vortex limits to L^-vorticity solutions.

Findings

Solutions can be obtained as limits of random point vortices.

The approach extends to initial measures with bounded densities.

The method generalizes previous Gaussian measure results.

Abstract

The 2D Euler equations with random initial condition distributed as a certain Gaussian measure are considered. The theory developed by S. Albeverio and A.-B. Cruzeiro is revisited, following the approach of weak vorticity formulation. A solution is constructed as a limit of random point vortices. This allows to prove that it is also limit of L^\infty-vorticity solutions. The result is generalized to initial measures that have a continuous bounded density with respect to the original Gaussian measure.