Almost global existence for the stochastic Navier-Stokes equations with small $H^{1/2}$ data

TL;DR

This paper proves that solutions to the stochastic Navier-Stokes equations with small initial data in $H^{1/2}$ exist globally with high probability, extending understanding of stochastic fluid dynamics under minimal regularity assumptions.

Contribution

It establishes almost global existence results for stochastic Navier-Stokes equations with initial data in $H^{1/2}$, a low regularity space, under smallness conditions on data and noise.

Findings

Solutions exist globally with high probability for small initial data.

Global existence holds on small time intervals without small noise.

Probabilistic estimates ensure near certainty of global solutions under smallness conditions.

Abstract

We address the global existence of solutions to the stochastic Navier-Stokes equations with multiplicative noise and with initial data in $H^{1/2}(\mathbb{T}^{3})$. We prove that the solution exists globally in time with probability arbitrarily close to~$1$ if the initial data and noise are sufficiently small. If the noise is not assumed to be small, then the solution is global on a sufficiently small deterministic time interval with probability arbitrarily close to~$1$.