The Annals of Probability

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

Nathan E. Glatt-Holtz and Vlad C. Vicol

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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

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 76B03: Existence, uniqueness, and regularity theory [See also 35Q35]

Keywords
Euler equations stochastic partial differential equations on Lebesgue spaces compactness methods pathwise solutions nonlinear multiplicative noise

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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