EXISTENCE AND UNIQUE WEAK SOLUTION FOR THREE-DIMENSIONAL STOCHASTIC GLOBALLY MODIFIED NAVIER-STOKES EQUATIONS ON UNBOUNDED DOMAIN

Abstract

In 2006, Caraballo, Kloeden and Real proposed a three dimensional alpha-model of Navier-Stokes equations in which the nonlinear term included a cut off factor  based on the norm of the gradient of the solution in the whole domain. The authors called this system the three dimensional system of globally modifed Navier-Stokes equations. Stochastic partial differential equations are a powerful tool to understand and study the mathematics of hydrodynamic and turbulence theory. To model turbulent fluids, mathematicians often use stochastic equations obtained from adding a noise term in the dynamical equations of the fluids. In this paper, we study the three dimensional globally modifed Navier-Stokes equations driven by additive white noise on some unbounded domains satisfying the Poincaré inequality. By the Ornstein-Uhlenbeck process, we transfer the stochastic system into a deterministic one with random parameters. Then, we prove the existence and unique weak solution for this system by using the Galerkin method.