## The Annals of Probability

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

#### 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 suitable nonlinear multiplicative noise. In the two-dimensional case we obtain the global existence of these solutions with additive or linear-multiplicative noise. Finally, 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.

#### Article information

Source
Ann. Probab., Volume 42, Number 1 (2014), 80-145.

Dates
First available in Project Euclid: 9 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1389278521

Digital Object Identifier
doi:10.1214/12-AOP773

Mathematical Reviews number (MathSciNet)
MR3161482

Zentralblatt MATH identifier
1304.35545

#### Citation

Glatt-Holtz, Nathan E.; Vicol, Vlad C. Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise. Ann. Probab. 42 (2014), no. 1, 80--145. doi:10.1214/12-AOP773. https://projecteuclid.org/euclid.aop/1389278521

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