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Global existence of the stochastic Navier-Stokes equations in $L^3$ with small data
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abstract
We address the global-in-time existence and pathwise uniqueness of solutions for the stochastic incompressible Navier-Stokes equations with a multiplicative noise on the three-dimensional torus. Under natural smallness conditions on the noise, we prove the almost global existence result for small $L^{3}$ data. Namely, we show that for data sufficiently small, there exists a global-in-time strong $L^{3}$ solution in a space of probability arbitrarily close to~$1$.
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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.
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