We consider the stochastic Navier-Stokes equations forced by a multiplicative white noise on a bounded domain in space dimensions two and three. We establish the local existence and uniqueness of strong or pathwise solutions when the initial data takes values in H¹. In the two-dimensional case, we show that these solutions exist for all time. The proof is based on finite-dimensional approximations, decomposition into high and low modes and pairwise comparison techniques.
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Glatt-Holtz et al. (2009) studied this question.
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