Small H^{1/2} initial data and small multiplicative noise give global-in-time stochastic Navier-Stokes solutions with probability arbitrarily close to 1 on the three-dimensional torus.
Local existence of the stochastic Navier-Stokes equations in the whole space
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We address the local well-posedness for the stochastic Navier-Stokes system with multiplicative cylindrical noise in the whole space. More specifically, we prove that there exists a unique local strong solution to the system in $L^p(\mathbb{R}^3)$ for $p>3$.
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Almost global existence for the stochastic Navier-Stokes equations with small $H^{1/2}$ data
Small H^{1/2} initial data and small multiplicative noise give global-in-time stochastic Navier-Stokes solutions with probability arbitrarily close to 1 on the three-dimensional torus.