On a Stochastic Leray-{\alpha} model of Euler equations

TL;DR

This paper introduces a stochastic version of the Leray-{ extalpha} model for 3D Euler equations, proving existence and uniqueness of solutions with finite energy under random perturbations.

Contribution

It establishes the first well-posedness results for the stochastic Leray-{ extalpha} model in 3D Euler equations.

Findings

Existence of a unique global solution in law for the stochastic model.

Solutions preserve energy formally despite randomness.

Results extend to 2D case with similar properties.

Abstract

We deal with the 3D inviscid Leray-{\alpha} model. The well posedness for this problem is not known; by adding a random perturbation we prove that there exists a unique (in law) global solution. The random forcing term formally preserves conservation of energy. The result holds for initial velocity of finite energy and the solution has finite energy a.s.. These results are easily extended to the 2D case.