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Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise. (English) Zbl 1304.35545

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 are considered. In the two-dimensional case the authors obtain the global existence of these solutions with additive or linear-multiplicative noise. They show that, in the three-dimensional case, the addition of linear multiplicative noise provides a regularizing effect; the global existence of solution occurs with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large.

MSC:

35Q35 PDEs in connection with fluid mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35Q31 Euler equations
35B65 Smoothness and regularity of solutions to PDEs
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