Existence of a local smooth solution in probability to the stochastic Euler equations in R3

We establish the existence of a local smooth solution of the stochastic Euler equations in R3. Probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data and the random noise. There are numerous works on the stochastic Euler equations in a two-dimensional domain. Even for the deterministic Euler equations in a three-dimensional domain, the only results are concerned with the local existence of smooth solutions. Our goal is to extend such local existence result to the stochastic equations. © 2009 Elsevier Inc. All rights reserved.