On local well-posedness of the stochastic incompressible density-dependent Euler equations

TL;DR

This paper establishes local well-posedness, including existence and uniqueness, for stochastic inhomogeneous incompressible Euler equations in three dimensions with both additive and multiplicative noise.

Contribution

It introduces a novel approach by reducing the stochastic PDE to a random problem and provides new estimates for transport equations in this context.

Findings

Proved local existence of solutions.

Established pathwise uniqueness.

Handled both additive and multiplicative noise cases.

Abstract

In this paper we study the stochastic inhomogeneous incompressible Euler equations in the whole space $\RR^3$. We prove the existence and pathwise uniqueness of local solutions with both additive and multiplicative stochastic noise. Our approach is based on reducing our problem to a random problem and some estimations for type transport equations.