On a 2D stochastic Euler equation of transport type: existence and geometric formulation

TL;DR

This paper establishes the weak existence of solutions for 2D stochastic Euler and Navier-Stokes equations with boundary conditions, demonstrating solutions are $H^1$ regular and of transport type.

Contribution

It provides a novel proof of weak existence for stochastic Euler equations with boundary conditions, including geometric formulation insights.

Findings

Existence of $H^1$ regular solutions under stochastic perturbations.

Applicability to bounded domains with Dirichlet and periodic boundary conditions.

Solutions are of transport type, extending classical results to stochastic settings.

Abstract

We prove weak existence of Euler equation (or Navier-Stokes equation) perturbed by a multiplicative noise on bounded domains of $\mathbb R^2$ with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are $H^1$ regular. The equations are of transport type.