Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise

TL;DR

This paper proves local existence of solutions for 3D stochastic Euler equations with general noise and global existence in 2D, also showing noise can regularize solutions in 3D under certain conditions.

Contribution

It establishes local solutions in 3D with general noise and global solutions in 2D, revealing noise's regularizing effect in 3D Euler equations.

Findings

Local existence of solutions in 3D with nonlinear multiplicative noise

Global existence of solutions in 2D with additive or linear-multiplicative noise

Noise can regularize solutions in 3D under certain conditions

Abstract

We establish the local existence of pathwise solutions for the stochastic Euler equations in a three-dimensional bounded domain with slip boundary conditions and a very general nonlinear multiplicative noise. In the two-dimensional case we obtain the global existence of these solutions with additive or linear-multiplicative noise. Lastly, we show that, in the three dimensional case, the addition of linear multiplicative noise provides a regularizing effect; the global existence of solutions occurs with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large.