Abstract
We consider the Navier–Stokes equations in \({\mathbb {R}}^d\) (\(d=2,3\)) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so Itô calculus cannot be applied in the space of finite energy vector fields. We prove existence of weak solutions for \(d=2,3\) and pathwise uniqueness for \(d=2\).
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Acknowledgments
Part of this research started while Z. Brzeźniak was visiting the Department of Mathematics of the University of Pavia and was partially supported by the GNAMPA-INDAM project “Regolarità e dissipazione in fluidodinamica” and the PRIN 2010–2011; he would like to thank the hospitality of the Department.
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Dedicated to Professor Bolesław Szafirski on his 80th birthday.
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Brzeźniak, Z., Ferrario, B. A note on stochastic Navier–Stokes equations with not regular multiplicative noise. Stoch PDE: Anal Comp 5, 53–80 (2017). https://doi.org/10.1007/s40072-016-0081-2
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DOI: https://doi.org/10.1007/s40072-016-0081-2