A large deviations principle for stochastic flows of viscous fluids

Abstract

We study the well-posedness of a stochastic differential equation on the two dimensional torus T2, driven by an infinite dimensional Wiener process with drift in the Sobolev space L2(0,T;H1(T2)). The solution corresponds to a stochastic Lagrangian flow in the sense of DiPerna Lions. By taking into account that the motion of a viscous incompressible fluid on the torus can be described through a suitable stochastic differential equation of the previous type, we study the inviscid limit. By establishing a large deviations principle, we show that, as the viscosity goes to zero, the Lagrangian stochastic Navier–Stokes flow approaches the Euler deterministic Lagrangian flow with an exponential rate function.