Considering the stochastic Navier-Stokes system in R d forced by a multiplicative white noise, we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical spaceḂ d p −1 p,r (R d ). The proof is based on the contraction mapping principle, stopping time and stochastic estimates. Then we prove the global existence of strong solutions in probability if the initial data are sufficiently small, which contain a class of highly oscillating "large" data.
The stochastic 3D Boussinesq equations with additive noise are considered. We prove the local existence of the strong solution in H s (1 2 < s ≤ 1). We also obtain a new stopping time, which shows that the H 1 2 + norm of u controls the breakdown of the strong solution. Furthermore, we give the probability estimate of the lifespan larger than δ (0 < δ < 1).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.