Sufficient Conditions for the Energy Balance for the Stochastic Incompressible Euler Equations with Additive Noise in two Space Dimensions

TL;DR

This paper establishes conditions under which solutions to the stochastic 2D incompressible Euler equations with additive noise satisfy an exact energy balance in mean, linking it to the tightness of Navier-Stokes approximations.

Contribution

It identifies necessary and sufficient conditions for energy balance in mean for stochastic Euler solutions, connecting it to the tightness of Navier-Stokes solutions.

Findings

Energy balance in mean is achieved under specific conditions.

Tightness of Navier-Stokes laws is equivalent to energy balance.

Numerical simulations support theoretical results.

Abstract

We consider vanishing viscosity approximations to solutions of the stochastic incompressible Euler equations in two space dimensions with additive noise. We identify sufficient and necessary conditions under which martingale solutions of the stochastic Euler equations satisfy an exact energy balance in mean. We find that the tightness of the laws of the approximating sequence of solutions of the stochastic Navier-Stokes equations in $L^2([0,T]\times D)$ is equivalent to the limiting martingale solution satisfying an energy balance in mean. Numerical simulations illustrate the theoretical findings.