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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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.

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

First available in Project Euclid: 9 January 2014

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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]

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


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.

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